I have three other options, but increasing your budget is probably your best bet.
1) Plan for an upgrade. Go with an Intel 1155 socket and go for a Pentium G2020 ($68 on Newegg). When you get the cash, upgrade to an Ivy Bridge I3.
2) Wait for AMD Steamroller APU's to come out in a couple weeks. They're rumored to be pretty good. Will they be? Who knows. It'll be about a month before you can get your hands on the parts, but it may be worth the wait.
3) Go with an AMD A10-5800K for around $120 or A10-6800K for $140
There's downsides to all of these options.
For option 1, you'd be buying a platform that is older and you'd need to ensure that you will be able to afford the upgrade in the future. Pentium dual cores were favored on tomshardware for a long while, but the recommendation was changed to CPUs that handle 4 threads for better frame consistency.
For option 2, you might wait a month and find out that steamroller is no good. Steamroller is RUMORED to be around Intel's sandy bridge with around a Radeon 7750 as integrated graphics. It will support AMD's new graphics api Mantle which will help in games codedd to support it. It's a new technology and a new processor generation though so there may have minor bugs for a while in drivers due to it's newness.
For option 3, this is the route I would avoid (due to my personal lack of knowledge on the topic). For a long while, there was consistency issues (frame pacing) when using APUs and the APUs didn't crossfire with other cards very well at all (performance was essentially limited by that of the APU alone). I don't know if AMD ever got around to fixing the issues or not. I think they fixed the frame pacing on the APU alone, but not the crossfire issues - I haven't read the latest on it. If you were to get a FM2+ motherboard, you could always upgrade to a steamroller chip later, but it costs upgrading costs more in the long run.