First, you're supposed to have an empty slot between graphics cards. If the images I'm finding for your motherboard are correct, you should have installed the cards in the first and fifth slots, which would put two empty slots between them (empty of full sized cards, anyway - something like a sound card that wouldn't obstruct the blower input would be fine). If they were installed in the first and third slots, the overheating is the fault of whoever installed them.
That said, if I didn't find the right motherboard, or there's some other reason they can't be separated properly, then you really do need a single-card replacement. That's no trivial matter, as a 5970 has a lot of rendering power. There are other factors to consider, however:
1) Four GPU's do not scale especially well in almost all games. Two GPU's scale very well - often perfectly. A third won't scale perfectly but rarely, but equally rarely fails to add significantly. The best option would be to replace your four GPU's with two GPU's.
2) High resolutions, especially with AA enabled, required more graphics memory. Given the price you mention, you have 1GB 5970 models (they'll say 2GB, but it's really 1GB per GPU, which is the number that matters - each 1GB must store the same data). Many games at 2560x1600 will be crippled by having only 1GB of graphics memory to work with. Crysis, for example. When I moved from two 4870x2 cards (four 4870 GPU's) to a single 4GB 5970 card (2GB per GPU and real 5870 clock speeds - $1100 card at the time), I was making a lateral move in raw processing power - the 5870 GPU is almost exactly twice as fast as a 4870 GPU. When I benchmarked the twin 4870x2 cards at 2560x1600 with no AA, it averaged about 34fps. The single 5970 averaged about 40fps.. Turn on 4xAA, however, and the 4870 setup with only 1GB of graphics memory turned into a slideshow, averaging between 10 and 20fps, with frequent stops at 0fps (having to swap textures in and out due to memory being at capacity). The 5970 dropped to 33fps average, with no slideshow effect. I've since encountered many newer games which use more than 1GB of graphics memory at the highest detail settings, even without MSAA. Tomb Raider, for example, uses well over 2GB with TressFX enabled.
I would not say anything higher than a 7850, as mentioned above, would do. A single 7970 GPU is somewhat slower than the pair of 5870 GPU's in a 5970 card (even underclocked). However, two 7970's would outperform two 5970's almost always. And at 3GB per GPU, you'll be well covered for gaming at 2560x1600.
As it happens, there are cards with two 7970 GPU's. The AMD-designed 7990 boards that recently entered the market are the ones you want - dual-slot designs with three cooling fans. At around $1000 to $1100, they meet your price point.
Regarding the addition of monitors, the first thing you should understand is that anything beyond two must be DisplayPort. That's true of a 5970 and a 7990.
The standard 7990 design has one dual-link DVI port and four mini-DisplayPorts.
It's not clear what you actually have right now, other than at least one 30" monitor. That will work with the DVI port. If you have two, and both require DVI, then you'll need an active dual-link DVI to DP adapter, which costs around $100 these days. A smaller monitor would work with a passive adapter, but there aren't enough pins to do dual-link passthrough.
Adding an HDTV via HDMI would be possible with a passive adapter, provided it's the only passively-adapted device connected. There are also active HDMI to DisplayPort adapters (around $40 is what I see with a quick check).
So if you have only the one 30" monitor right now, you should make sure any additional monitors support DisplayPort natively. I have a pair of 7970's myself, and have an HP LP3065 connected via DVI, and an HP ZR30w connected via DisplayPort. I've previously had the LP3065 connected via an active adapter to the 4GB 5970 (which had only six mini-DP outputs).
Long answer, but I tried to be reasonably thorough. Good luck.