As mentioned, with mixer 'boards', you need to read the fine-print to see if it has multi-channel usb capabilities. Often times, though they have multiple channels on the mixer itself, they may only have a stereo mains input to the PC. I just the other day was looking for an audio interface and was considering a 'board' as a solution (I need mixer and FX capabilities), but dropped it once I realized that it could be an issue.
I went to the used market eBay; managed to find a new-unused (opened box) Motu Ultralite MK3 just at the very outer limit of your budget, but others, maybe not quite so new, have gone for even less. They have two pre's plus six more balanced ins, 8 balanced outs, all inputs and outputs are available at the DAW (USB or FW) , and it functions as a mixer (good for tracking) as well.
Getting one of these for a reasonable price on eBay is doable if your willing to sit on an auction in its last few minutes. It's a pain to do, though in my case, available fundage necessitated it...