Hi and welcome to POTN...
Your lenses will work fine, as stated... but - as some others have already noted - will be subject to the "crop factor" with most cameras at this price level. However, you don't really need to rush out and get another telephoto zoom at this time. I would note, though, that there are several different 75-300 and 80-200 models of varying capability/image quality... depending upon which particular version you have. You may or may not want a better lens eventually. But for now, I'd suggest simply getting a camera with a wider (shorter focal length) walk-around lens for use with the camera. Bought in a kit, this can be relatively inexpensive.
The most common kit lenses are 18-55 IS and 18-135 IS... the latest versions of each of these have a newer focusing system called "STM", but of the cameras being considered only the latest T4i (650D outside of N. America) really takes full advantage of them. You might find some kits, particularly with T2i or T3i, include the earlier non-STM versions of these lenses, which should cost a little less. They will likely be a little slower focusing, but still quite usable. The kit with 18-55 will be cheaper than a kit with 18-135mm... but the latter will be more versatile. Either of these kit lenses would complement your telephoto zoom nicely and give you a very versatile kit.
Coming from a film camera, 18mm may sound super wide angle... but it's really not on these "crop sensor" digital cameras. It will give an angle of view that's roughly equal to a 29mm lens on your film camera. Multiply any focal length by 1.6X, to see how it would seem on your film camera. So, for example, the 18-55 would be like 29-88mm and the 18-135mm would seem like 29-216mm.... and, as noted above, your two telephoto zooms will both seem like much stronger telephotos as well.
As noted, among current models and buying new you have choice of T3, T2i, T3i, T4i and 60D. Within your budget, you'll be fortunate to find T4i with a single kit lens. 60D you might have trouble finding with any lens at all, or even within your budget (though you might look for used or at the Canon Direct Store website for refurbs, which might come with some limited warranty).
Sure, a used 5D classic is an option, too. Full frame will "feel" more familiar with your current lenses and it can be found within your price range. Just keep in mind that it's a model that was introduced in 2005 and replaced by the Mark II in 2008... So is now a few generations old technology. Still, 5DC are quite capable. If you consider buying one, try to be careful about getting one with "too high mileage". There was an issue with the mirror coming unglued on the model, though Canon will still repair that for free, last time I checked. As to any other repairs, it's hard to say how much longer Canon will support the model with spare parts and offer in-house repairs (though it might be possible to get repairs via an independent shop for a while longer).
Canon's software no longer supports the 10D, Digital Rebel/300D and earlier models (both those models were intro'd in 2003 and shoot "CRW" RAW files). For now, they still support the 12 bit "CR2" RAW files of 20D (2004), Rebel XT/350D (2005) and 5D classic (2005)... But it's lilkely just a matter of time until they only support 14 bit "CR2" files of 40D (2007), Rebel XTi/400D (2007), 5D Mark II (2008) and later. Of course, third party softwares such as Adobe Elements, Lightroom and Photoshop continue to support all the earlier file formats without issues.
It wasn't something we worried about very much back in the days of film, buying the "latest and greatest". The 35mm film SLR was pretty well sorted out by the 1970s and only relatively minor incremental feature improvements took place from around 1980 and onward. They all handled the same films and lenses were the biggest differentiator by far.
Now with DSLRs there have been much more rapid and dramatic advancements... my 10D bought new in 2004 is still a great little camera, but seems rather primitive by today's standards! There have been 5 more generations of xxD cameras in the 8.5 years since the 10D was intro'd. We've gone from 6MP 12 bit image files to 18MP 14 bit. A lot has changed and DSLRs have rapidly improved, though they are still just cameras at heart! I bought and used film SLRs for one or two decades, where more recently DSLRs have sometimes seemed pretty dated within just two or three years (Thankfully, I think this is slowing down now and DSLRs have "plateaued" in development, to some degree.)
If you prefer to buy a new model, again, within your budget you'll be limited to the T3/1100D (the most entry level model currently), T2i/550D, T3I/600D, T4i/650D and 60D. The last three have an articulated rear LCD screen, which might be a nice feature for video or low/high angle still shots. All except the T3 share basically the same 18MP 14 bit sensor, so there's practically no difference in image quality between them. The T4i and 60D share a little bit more sophisticated AF system, 9-points with all of them the "better" dual-axis type. All the other models have a 9-point system, with only the center point the dual-axis type and the other 8 single-axis. The Rebel/xxxD series use a "penta-mirror" based viewfinder, that allows them to be a little smaller, lighter and less expensive. The 60D has a true pentaprism that makes for a slightly larger and brighter viewfinder. 60D also has 1/8000 top shutter speed and 1/250 flash sync, while the Rebel series have 1/4000 tops and 1/200 flash sync.
The control layout of 60D is a bit different than the Rebel series models... (7D, 5Dxxx and 40/50D are different from either). I'd suggest going to a store and checking them out, comparing hands-on and in person, to see if you have a preference one way or another. There's some difference in size and ergonomics, too.
Have fun shopping!
