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Sounds like a high resistance connection somewhere. I would start by checking grounds or maybe performing a continuity test. Your blower motor could be dragging too.
I am having the same problem. Blower motor resistor sells for 6.29 at Advanced auto parts. That is the first thing I am going to try. I found a good write up and diagram here... http://www.justanswer.com/questions/...004-dodge-neon
If I can remember, I will let you know if it works.