You would need a new motherboard anyway, I say go with the FX 6300 and Am3+ motherboard. The 6300 is a HUGE upgrade over Q8300 and a big upgrade over any i3. If your budget is that low, you can't go wrong with AMD. The FX 6300 @ $119 is better than every processor any where near it's price range.
To answer your original question, I think the i3 3220 would be superior to the old q8300 in gaming. I know for a fact that a i3 2120 is strong enough to produce 60+ fps when paired with a GTX 660ti.
If you cannot afford an i5, go with a FX 6300