Could I add a potentiometer to a dremel?
Adding a resistor will make the speed regulation with load crappy. You want to actually reduce the voltage, either by a variac or by some kind of phase control, like the variable speed Dremels have. 

You got a good suggestion to check the brushes (you don't even have to open the case and they're something like $3 for new ones), and if the speed control (assuming you have one) is faulty, it's about $25 for a new one. 

If you do decide to replace the tool entirely, you might want to consider a Foredom.