I just uncovered that several outlets in my house were wired with copper wire running from the ground to the neutral. From everything I've read this appears to be the result of shoddy work and is quite dangerous. I don't have a lot of money to spend on the problem but want to make sure I make the house safe for myself and future owners. What are my options when it comes to correcting the problem? I really don't want to run new wiring but haven't found any reasonably priced alternatives.

Any neutral/ground connection does need to be disconnected.

You can replaced any outlet grounded to the neutral with a GFI receptacle, if you can figure out the circuitry, one GFI device at the beginning of the circuit can protect the remaining device downstream.

Any outlet without a ground gets one of the labels that come with GFI receptacles "No Equipment Ground".

Or you can install a GFI breaker, you still need the labels, but none come with the breakers, if I am not mistaken.

Rewiring with new cable that has a ground is the best, but can be costly.

Those are the only methods I recommend.