First of all, I would not recommend someone wanting to shoot sports jump to a full frame model. In order to get the same telephoto "reach" with FF, you will need to spend a lot more on lenses and your kit will be larger and heavier. The 6D is sort of like a FF 60D, with a less capable AF system and without the articulated LCD of the crop model. Plus for 8x10s you don't need FF at all. In your position, I wouldn't even consider a FF model.
I do not know the T3's high ISO performance and you don't indicate what ISOs you have run into trouble with... But you can reliably use any of the 18MP models (T3i, T4i, T5i, 60D, 7D) to ISO 1600 and even 3200 with minimal work. Especially if you are rarely printing larger than 8x10. The 70D has a new sensor and might offer even nicer high ISO performance... Canon says so, but I'm still in wait and see mode, personally.
You can use even higher ISOs, by applying noise reduction in post-processing. There have been significant advances in softwares the past few years, allowing for higher ISO shots. Look for Teamspeed's posts here on POTN about using 7D at very high ISOs for examples and recommendations. The same techniques should apply to any of the 18MP cameras (don't know about the new sensor in the 70D yet).
7D is the most advanced of the current models... largest buffer for fast shooting, sophisticated and customizeable AF system. It's also got the highest durability rating of any of the Canon APS-C cameras and relatively good sealing against dust and moisture.
The 70D comes close and has some features the 7D doesn't. But 70D is more of an upgrade from 60D, with some 7D features added. In particular, the AF system of 70D is very similar to 7D's, but slightly downgraded. And it doesn't offer the 7D's build level of sealing and durability or 100% viewfinder. But it does offer an articulated LCD screen, which the 7D doesn't. 70D also has a newer, more sophisticated form of focus Micro Adjust than 7D (60D doesn't have MA at all).
The 60D's 9-point AF system is perfectly capable of action photography, too. It's a solid step up from your T3's AF. In fact, the T4i and T5i models have essentially the same AF system as the 60D, so should be able to keep up, too.
High frame rate is a crutch you want to avoid using too much, unless you have a ton of memory cards and enjoy sitting in front of your computer for days on end editing your photos. Learn to time your shots well, rather than relying too heavily on "machine gun" shooting techniques. 7D has the highest frame rate, but 60d and 70D are pretty respectible, too.
The way I most often use the 8 FPS my pair of 7Ds are capable of is with short, 2 or 3 image bursts. Usually this is with running subjects where a "freeze frame" of the wrong part of a stride can look odd or awkward, and it's near impossible to time. This is sort of like shooting a gun using a "double tap" or "triple tap" technique. You end up trashing a lot more images this way, but increase the odds of stopping the action at a point in the person's or animal's stride that looks "right".
Besides, 7D will actually slow down the frame rate at times, to allow time for metering or AF, or with certain settings. To get a 7D to fairly consistently shoot at it's max frame rate, you have to use M (manual) exposure mode and keep some other features turned off. Plus there really isn't all that big difference between 5, 6 and 8 FPS.
I would encourage you to spend as little as possible on the camera body.... you also sorely need to upgrade lenses and those will ultimately offer more improvement in image quality, as well as support faster, more accurate AF, better sealing and durability.
You should look at the Canon 70-200s, especially. This will be a huge improvement over the 75-300 you have. Some will tell you that you don't need IS for sports photography, but after using IS lenses for 12 years I can tell you it's a very valuable feature on telephoto lenses, especially when they are used on crop cameras. If a 70-200/2.8 IS Mark II is too expensive, look for a used Mark I or consider a 70-200/4 IS. They are both excellent lenses, too. They can be used very effectively with a quality 1.4X, too, if you need a bit more reach.
Besides a "workhorse" 70-200, it's your choice what other lenses to get. I use a 24-70/2.8 and a couple 28-135 IS. The Canon 17-55/2.8 IS is another excellent lens. An inexpensive choice is the Tamron 17-50/2.8 non-VC.... except it doesn't have USM style focus (which Tamron calls USD), so is rather slow focusing. The Sigma 17-50/2.8 OS is in between in price and has HSM, which is the Siggy version of USM, to give AF speed and accuracy similar to the Canon.
There are numerous other short zooms, but most are f3.5-5.6 variable aperture, so are going to be less useful in low light situations. Instead you might consider some faster prime lenses, such as the Canon 28/1.8, 35/2 IS and 50/1.4 (all with USM) or the Sigma 30/1.4 HSM (tho it's said to not be so great focusing in low light). A prime lens can give you one or two stops more light than any zoom offers, yet can cost less and is smaller/lighter, too boot. Of course, a prime isn't as convenient as a zoom in some respects.
Note that with very wide and ultrawide lenses you may not need a large aperture lens because you can handhold them steady at slower shutter speeds. Often an f4 or f3.5-5.6 will do fine, in a 10-xx or 12-xx zoom.
You also might want a longer telephoto for sports. I use the Canon 300/4 IS a lot, both with and without a 1.4X teleconverter. It's easily handheld and a nice focal length on a crop camera for field sports. There is also the Canon 400/5.6, but it might be a bit long some of the time and doesn't have IS, though it's reasonably compact and handholdable. There are also a number of zooms in the 100-400, 120-400, etc. range.
Anyway, I recommend you keep as much of your budget as possible to put toward lenses. I'd much rather get a 50D, 60D or T4i body and have a good selection of quality lenses to use upon it, than the latest and greatest camera body with cheaper lenses. The lenses make more difference than the body, in your images and even can help with AF speed and accuracy.