Well 300w PSU really isn't enough. Most people will recommend a decent Socket 939 motherboard and CPU which totalled up will cost around $300 for those two items, possibly a little less.
But without a better PSU I wouldn't be surprised if you had the same problems.
I'd suggest looking into a 350-400w with stronger rails, mainly the +12v to ensure the CPU, fans, and other major components get plenty of power.
This is especially true if you've put the components into another computer and they work fine. You could always swap out the mobos to truly find out if that is your problem, but obviously that'd be somewhat of a pain in the butt.
ABit AV8-3rd Eye
are both good Socket A boards, BUT many will argue that you should go the PCI-E route, but since you'd like to reuse your AGP video card and you're doing business apps there's really no need to.
AMD64 3200-3500+ should be adequate for your CPU needs.