If you get a new mobo you will have to get a new processor also I think. Just keep the mobo and get an adapter. You will see only marginal gains at best with a new mobo. It's not worth it.
Anyways the only difference between the 20 pin and 24 pin connector is that the 24 pin is used so the mobo can control the speed of the fans in the PSU.
A 24-20 pin adapter works fine ive installed them on several computers and never had a problem. Stick with getting the adapter if you dont want to spend a lot of money because you wont want to spend much but then in the long run you will realize that you talked yourself going over your budget LOL it happens to everyone.
Best to just fork over the few dollars for an adapter.