I have no idea the prices in your area, but around here, 7D are on sale big time... probably in anticipation of 7D Mark II. You can pick up a lightly used or refurbished 7D for about $750 US. New are going for around $1000. Sell your 5DIII and get a pair of those. 7D is sort of a "1D Lite"... has similar dual processor, separate AF chip arrangement, shutter release, higher durability shutter (150K rated) and 8fps frame rate. Upgraded build and sealing, too. 100% viewfinder. 18MP, (compared to 10MP 1DIII or 16MP 1D IV.If you like the vertical grip of 1D-series, add a BG-E7 to the 7D. Those can be found for around $150 US new at present (Note: 7D cost $1600 US and BG-E7 cost $250 US five years ago, when I bought my pair of them). If you can't get publishable images with 7D.... well you can't get them with anything. I'm sure you know this, since you've already used one.
But I really don't think the camera bodies you choose matter very much. You have no lenses to speak of, and those are far more important. Just get whatever camera(s) leave enough in your budget to add lenses, too. Sure, a 1D series camera would be great... but I wouldn't hestitate to use 70D, 60D or even 50D. I also don't think high ISO capabilities are all that big a deal... buy a flash.
But mainly, you need lenses!
Keep your 24-105 for now. Eventually you may want to replace it with a 24-70/2.8. Or maybe not.
You really need a fairly strong telephoto for sports. I'd recommend a Sigma 120-400 OS as the most affordable that will cover a great range for sports shooting. It's an f4.5-5.6 lens, but that won't be a problem in most reasonable daylight situations. I use my 7Ds up to ISO 6400 (shooting RAW, avoiding underexposure and applying some Noiseware noise reduction during post processing), which allows fast enough shutter speeds for sports at f4.5-5.6 even in heavily shaded/indoor situations. The Canon 100-400mm is an alternative, but much more expensive, whether new or used.
Alternatively, you might be able to find a third party or used 70-200/2.8 within your budget... Canon "Mark I" with IS are still excellent, if you can find one at the right price. Sigma and Tamron both offer them, though I am not that familiar with theirs or the pricing. For that matter, Canon 70-200/4 IS also is excellent and may be fast enough for you. I use one quite a bit, as well as an f2.8 IS "Mark I". The problem with 70-200 is that it's not anywhere near long enough for most field sports and some other things where you have to work from a distance, such as motorsports, cycling. You can add a 1.4X to a 70-200, but there's some loss of image quality. Instead I often have a Canon 300/4 IS on my second camera, though. I sometimes use that with a 1.4X.
Even if you keep your 5DIII and work with it, you need longer lenses, especially if you hope to shoot sports. With the full frame camera, you'll need longer telephotos than you would with and APS-C or APS-H.
You also might find some use for a wider lens. This is one place that switching to a 7D or other APS-C/1.6X crop camera can really help with your budget. There are a lot of choices of good ultrawide zooms for croppers, at reasonable new and used prices. In general, a high quality wide angle zoom for full frame or APS-H/1.3X tend to be much more expensive. You can pick up a new APS-C ultrawide in the 10-20 or 12-24 range for around $500 US. If you buy used, you can probably find something good for $400 US or less. In comparison, FF/APS-H compatible wide angles are mostly $1000 + lenses.
Can't help with your uploading issue... You'll have that problem with any fairly modern camera and the file sizes they can produce. In all cases you can shoot RAW + less-than-largest-size JPEG. Ask the service you're submitting to, what they recommend and their minimum file size. If you could send via a wired connection, it may be faster.