You can have your heat pump converted to an oil hot water coil unit. There are several makers of boilers that sit outdoors like your heat pump that do not require a chimney. Axeman-Andersen, Triangle-Teledyne, Burnham-New Yorker
There are also several that can fit in a closet and use direct venting to the outside. Triangle, MHI, Ultimate, System 2000
The new boiler will be able to heat the home through adding a hot water coil to the warm air system. In addition, you can add baseboard or additional blowers to rooms that need more heat. The boiler can also make more hot water than you will need for 1/3 the cost of the electric water heater and save $600 per year on the cost at the same time. The oil boiler will pay for the conversion with the electric bill savings and make you much more comfortable.
The water drain is not a great problem. Any good heat pump installation company can take the bottom of the pan off - it is only sheetmetal - and make a deep one that drains properly.
Part of the problem is that you are calling Sears to do a Carrier repair. Call the area Carrier representative and ask him for a reputable dealer.