Okay, I'll try to address some of your concerns now. To start off, the Canon 750D is the same camera as the T6i. I actually own this model of camera, among others in the Rebel series, so that should help.
I guess my question for you is, which types of photos are coming out less than stellar? Is it the landscape or action shots? To test your camera out, place it on a tripod and take a few photos (in auto mode) of some outside objects that are still. Mix up the lighting too. Take some on sunny days and some on cloudy days and then compare the images and specs. You can zoom in and out if you want, just to see what's happening. If everything comes out okay, then there's nothing wrong with your camera or lenses.
From your question, I suspect a few things may be occurring. They have to do with lighting and depth of field while taking action shots. I'm not sure how experienced you are as a photographer, but there are a few areas that can get even the best of us. And just to let you know, action photography and proper lighting is notoriously challenging. It takes a lot of practice and can get technical. But if you understand the basics of how the camera and lens works, you can easily get a grasp of the situation.
I'll go through your post point by point. I think that may make things easier.
I feel I'm losing performance with it.
I have never actually heard of a camera losing performance. That doesn't mean this type of thing doesn't exist. After reading what I have to say and testing your camera out, if you still have problems, you may need to take it to a shop.
I take quite a range of photographs, from landscapes, portrait to sports; mainly animal portraits, performance horses, and moto cross.
Let me guess; landscapes are generally still in focus without much, if any grain, portraits look good, as long as there is sufficient light, but if there isn't, you'll see some grain, performance horses and moto cross shots don't have much grain if the pictures are taken outside, but once you go into an arena or something like that and the horses are moving, you see grain. Whether you're inside or out, you are starting to see some blur in your distance action shots. Not motion blur, but overall blur. Please tell me if I'm correct.
I took photos at Royal Highland show 2019 and my photo quality was fantastic. I took photos in the main arena (from a spectators seat!) where I needed to zoom in and most photos were great quality.
The most important question I have for you is in regards to the lighting. Was this indoors or out? How was the lighting? I suspect there was lots of light. I'll explain why this matters so much below.
I always shoot in raw.
Shooting in RAW or JPEG would make no difference here. I think what you're describing is technical in nature, not how many pixels are contained in the final file. A photo shot in RAW has just as much chance of being grainy or blurry as a photo shot in JPEG does.
My lens is the 35-350mm L series and I love it, but lately my photos are showing out of focus or grainy.
This right here can be a huge problem if not handled correctly. You are using a lens that can be very difficult to handle if you're not in full manual mode. I'll explain more below.
To make sure it wasn’t just me, I went back to using sport mode on auto focus (what I used at highland show) and used manual focus too and both were not very great. I was photographing a show jumping event. No tripod, but I stood very still it was not windy and still my photos are poor quality. Do you think I would need to send my camera away or my lens to get checked by a professional? I have now got a few events to do and I'm not so confident my photo quality will be up to scratch. Also, I've noticed this camera does not like it when the weather is dull. Is there any way around that?
Okay, here goes. If you were sitting in the stands of an indoor arena with your 750D set to auto and with a 35-350mm lens attached to your camera, zoomed in all the way, a few things would occur. First, by zooming in all the way, your aperture would open up as wide as possible. The reason for this is to allow more light to pass through the lens. As you can imagine, by extending the tube of that lens as far as it could go, light has a very tough time passing through. The wide aperture would attempt to combat that darkness. The problem is, by opening up the aperture as much as possible, you're also reducing the depth of field of the scene. Depending on your distance from the subject, the depth of field can leave little room for clarity (non blur). So if you had a horse or a dirt bike that was moving towards or away from you, a novice photographer would have a nearly impossible time trying to keep that fast moving object in focus.
To compound this problem, I'm assuming you're either using sport mode or shutter priority to reduce motion blur. With a fast shutter speed and a long dark tube lens, there's little opportunity for light to pass through to your camera's sensor (even with the larger aperture, which probably isn't that large to begin with - most likely a maximum of 3.5 or 4.5). The only area that's left to compensate for the lack of light is the ISO setting. I suspect that if you looked at your specs for the images you don't like, the ISO would be very high. There's the cause of your grain.
So there are two things going on here. If you were to photograph a moving dirt bike that's coming towards you outside while your lens is zoomed in and with a fast shutter speed, you're dealing with the issue of a shallow depth of field. I suspect these are the shots that are out of focus. There's no motion blur because your shutter is set so fast, but there is an overall blur that makes you think the lens isn't focusing correctly.
The other problem is lighting. By zooming that long lens in all the way and by using a fast shutter speed, you're reducing the light that hits the sensor to such a degree that your ISO value is shooting up far too high in an attempt to achieve an exposure that's correct for the situation.
Have you ever seen wildlife photographers take photos of eagles and running bears? They always use those huge lenses. The reason those lenses are so large (and expensive) is because they've got huge apertures inside of them. It's all about letting light through the lens.
If you were to take the same types of action shots as you're currently taking, but to set a slow shutter speed of, say, one second, you'd most likely see a resulting photograph with little to no grain. You'd also see tons of motion blur. There would be tons of light hitting the sensor, but because the shutter is open for such a long time, there would be trails of horses and motorcycles.
Another limitation is the camera itself. Because it's got a cropped sensor, it can collect only a fraction of the light that a full sensor camera can collect. So in some respects, the camera and lens themselves are playing a part here. But because most of us are on a budget, this type of equipment is what we purchase because of affordability.
If what I'm saying here ends up being accurate, I have some advice. If you aren't in full light situations, get closer to your subject and try not to zoom in all the way. You may also want to leave this 35-350mm lens for still photography, such as landscape, where you can allow for a slower shutter speed. When it comes to action, this lens may not be the best choice. Something shorter and with a larger aperture may be more appropriate. Also, getting closer to the subject will probably help.
If you're dealing with action shots where a motorcycle or horse is moving left to right, you can generally autofocus on the moving object and track it with a fast shutter speed and get decent results. If the moving object is coming towards you though, you'll definitely want a smaller aperture to leave a deep depth of field so much more area stays in focus longer.
Please let me know if anything I said here makes sense to you. I have a feeling that your equipment is fine, but some of your settings and choices need to be modified.