Well, first off, unless your willing to get the new Nikon D800 (which has a 36.3mp sensor...and is supposed to list around $3000), its unlikely you'll find a cheap camera that is fully capable of reproducing at 300dpi (I assume you mean in final print here...its tough to correlate scanner DPI to digital photo resolution), let alone at the quality you could get from a scanner.
For the highest resolution APS-C sensors, which are probably Canon's at 18mp (Sony's 24mp APS-C sensor suffers from pretty high noise, not to mention the fact that it likely outresolves lenses by a fair degree so the added resolution is pretty moot), your going to have less quality than would be ideal. You are unlikely to actually resolve the same level of detail as a scanner as well, as your packing the entire image into the area of the sensor, where as a scanner will use a high resolution line scanner across the whole page. Most older digital cameras have sensor resolutions that top out much lower than would be necessary to replicate your artwork in print at 300dpi for A3 size (16.54x11.69 inches, or 420x297mm). You could pick up a brand new Canon 550D or 600D for less than $1000 (around $600 for the 550D). The native print resolution for one of Canon's 18mp APS-C sensor cameras is 17.28x11.52 inches, which doesn't really leave you any room for crop/rotation (at least along the short edge.)
If you could find a used Canon 5D Mark II, you might have enough resolution. Its a full frame camera, so it manages noise extremely well (although that may not really be a problem if you light your artwork properly.) If you light your artwork well, slap the camera on a tripod and shoot with a cable release/teathered, you should be able to use ISO 100 to get clean images with barely enough resolution to support a little post-process crop and rotation (to clip off any unsightly edge factors and rotate to the proper orientation, etc.) for final print. The 5D II's native print size at 300ppi without any form of cropping/rotation would be 18.72x12.48 inches, so you would probably land square on the mark for native A3 size post-crop/rotate. The 5D II lists at $2700, and resales for pretty close to that most of the time. There have been some sales lately for brand new ones at $1999, but thats not really any different than spending $2000 on an A3 compatible flatbed anyway.
As far as quality goes, I can only speculate. I've seen some phenomenally good drum scans of medium and large format positives at 300dpi. The image resolution from such scans tends to be quite large, often much larger than a comparable digital sensor (even the new 36.3mp Nikon D800). I'm not sure exactly what the quality from your previous A3 flatbed scanner was, or what kind of optical resolution it actually supported. Most A4/Letter/Legal flatbed scanners I've used these days support at least 600dpi for "normal" scans, and sometimes up to 4800dpi or even 9600dpi for film scans. The quality you can get from a $600-$800 scanner is pretty darn good, all things being equal.
I would say that using something like the Canon EOS 550D would certainly produce adequate, usable results, even if they are not quite as high quality as a flatbed scanner designed for it. Canon cameras can be used "tethered" to a computer as well, such that the computer can be used to control the cameras shutter, and such that the photos are saved directly to the computer. This can be done with Canon's DPP (Digital Photo Professional) or Adobe Lightroom, both of which work on Mac. I'd recommend Lightroom 3.x or 4 beta myself, as you can easily catalog your work, add metadata, even print...as soon as the photo is taken, and otherwise handle your entire workflow in a single application. I would recommend getting some high resolution glass as well. At 18mp, an APS-C sensor resolves a LOT of detail, more than many of Canon's cheaper lenses can resolve themselves. Canon's newest lenses, releases from the last few years, primarily their Mark II lenses (i.e. the just-released 24-70mm f/2.8 L II) are designed to resolve enough detail for 18mp APS-C sensors or 46mp FF sensors. I would be very wary of using the native 18-55mm kit lens that comes with the 550D, as its pretty bottom of the barrel, and probably not sufficient to resolve maximum detail at 18mp. A prime lens would really be best, although they can be pretty expensive. There are some cheaper primes, such as the 50mm f/1.8 II, 100mm f/2.8 Macro, etc. that cost around $100-500 new, however I am not sure they are capable of fully resolving enough detail for 18mp (especially in the corners, where sharpness drops off quite a bit, despite the fact that hose lenses are considered very sharp. Compare the MTF charts for those two vs. the new 24-70mm f/2.8 L II which was explicitly designed to resolve high resolution.)
It may also be that your idea of quality and my idea of what you want for quality are very different. The kit lens may be entirely sufficient for your needs (especially given that its recently been updated, and the new 18-55mm lens might be considerably improved over its predecessors.) If you have any friends with the 550D or 600D with kit lens, I'd see if you could borrow it and give it a try before you put any money into lenses. It may also be that to get enough detail out of the body+lens you would still have to spend over $1000, in which case its probably worth spending $1500 on a new scanner anyway, as its likely to do a better job for this particular task.