FX 8350 owner here.
Seen how you have to get a new board anyways, why not go for a 4th GEN intel, like a 4590 or a 4690/k? They would be much much better than AMD in games. Your 2400's gaming performance would probably be very similar to an FX 8320/8350 anyways. To get decent performance outta an AMD Chip you have to overclock the crap out of it which means upping the voltage which means shortening the life of the chip and voiding your warranty.
It wouldn't honestly be worth swapping over to AMD at this time, if you had a 2nd gen i3 or something it might be worth it, But you have a capable i5.
I personally, would wait a year or so until the 2400s isn't performing up to scratch anymore then see what the new AMD's are like and how they stack up against the existing intel range.
You would not see a boost in anything if you swapped to a FX 6300.
Edit: Just read that you didnt want to spend that much on a 4th gen. In that case I'd see if you can find a second hand 3rd gen i5 like a 3570k seen how it would be compatible with your board.