Courtney, it actually sounds like you need a knowledgeable Roofer that will know how to find the source of the leak, weather it's with the roof membrane or perhaps the A/C coils dripping to the inside (as an example, it happens!). 

The oil burner we replaced was burning a gallon an hour when running and it ran 12 months a year due to domestic coil.

The trick is learning that number.I pay:Interior repaint (3 colors): $1.5 - $2Flooring: $1.5 Roofing: $300 (all in materials and labor no deck repair)Windows: $75 per openingHvac: Condenser/a coil = $3k furnace = 3k (high, but I get priority and zero upcharge ever)All new pex $1500That's a basic list, but you get the idea.

They found the unit had not been cleaned (dirt caked up on coils and the unit was low in freon indicating there is a leak in the system.  

I believe the aversion to electric is coming from the old style stoves with exposed electric coils.

There is a pump in the tub that circulates water through the coils of the heat exchanger. 

Of course they are not likely to clean the filter or rinse out the coils to keep it running efficiently, whether it is your or their AC.

I just put two new AC compressors and A-coils into two new rentals since the existings units were in pretty bad shape and I didn't want to worry about them going down soon. 

I would schedule a maintenance, clean the coils, blower, and give it a tune up since you are flipping.

Swinging a hammer is impossible in tight spaces and a regular nailer isn't precise enough to hit the holes in the various hangers.