I just got an AC quote from Sears and I was very tempted to report him to the State Attorney General for being a complete shyster! He was telling me the size unit I needed over the phone. He asked me "What kind of house do you have?", I said "Colonial" and he said "you need at least a 3.5 - 4 ton unit." Well, enough rant...
The condensation problem you have doesn't sound like a major big thing. The condensation needs to be collected, put in a pump, and ejected. If it is failing at one of these points, or needs more maintenance or reconfiguration then someone should be able to come in and fix it, done. I have a hard time believing that a malfunctioning condensation situation could be the cause of needing a new unit. This doesn't sound right. I'm not an HVAC pro but this sounds like a pile of dung to me.
Converting to oil might not be the way to go. You would need a chimney going up through the roof and it doesn't sound like you have one if you have a heat pump. Some gas furnaces have cool direct vent setups where you can just go out through the sidewall. This is a pretty big decision and a complicated one too. There are many factors to consider. I would ask all your friends if who their heating/cooling contractors are and how they like/trust them. Pick one, have him fix the condensation situation and make general recommendations/price quotes. More people will chime in on this whole question.
I think upstairs rooms tend to stay warmer, no matter how well ventilated, insulated etc.. Heat rises. Proper roof venting can help a lot but we need more details on what you have now. More insulation in the attic can help. Better or larger return air ducts upstairs can help. Running the A/C unit fan continuously can help by constantly moving the upstairs air down and vice verse. OR, you may need a second zone, ie: trolatemp or sim. ($1,000 +).
Duct cleaning, see "Duct Cleaning" in the yellow pages. OR, your return air and/or filter area may be deficient. How often do you replace your filter?
I would get the water situation fixed immediately. Get a dehumidifier into the basement and/or a big fan and lysol etc. The mildew could be very dangerous to your health!
My community has 4-5 main HVAC contractors that do most of the jobs and are very reputable. You need to ask around, not be sold a bill of goods and my advise, from my experience is RUN AWAY FROM SEARS!!! I'll probably catch flack for this, but what the heck!