I also live in NH. It is legal for renters to have a wood-stove as a primary or secondary source of heat, but the landlord has to approve it, which means having an insurance company inspect it so it can be covered in the insurance policy held by the property owner. This is something you'd probably have to reimburse, at least a portion, perhaps, in the rent.
Otherwise, if you're living in a house that gets water (mold danger), and you're venting a dryer inside (more moisture and mold danger), and the house is not well-insulated, plus it's costing an extra $650 per month for heating it in the winter, I'd suggest forgetting the audit and move. Seriously. It sounds like you'd save money in renting something safer and warmer rather than spending it on sketchy heating methods that aren't reliable. (What if we lose power, which is the rule in New England winters?)
It helps to seal drafts in order to maintain heat, but in your situation, I'd be worried about sealing in moisture and mold, at least. If you can't move presently, then an energy audit is the LEAST I would have done.