Other than the fact that your shifter may operate the opposite of how it does now, that derailleur should be fine.
You only need something from Shimano with "MegaRange" to handle the 34 tooth sprocket, since you are replacing a shimano indexed rear derailleur. (I am assuming indexed)
There are better rear derailleurs, like a Deore LX which will handle the 34 tooth and you could find one near the top end of your budget. icyclesusa.com has one on sale for just under $30 I think that is where I bought mine.
Whether or not the higher end RD will make a difference is the subject of many debates, but since you need to replace it anyway, I would consider going for the best you can afford... Of course that assumes you don't have to turn around to get a new cluster or chain if you find out that more was wrong than the RD. I would say that bigbossman probably looked and realized that the issue wasn't the cluster or chain, so you should be fine for some period of time.
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