Efficient whole house heating on a budget. Electric baseboard?
Electric baseboard is "100% efficient" - all the heat gets used in the house. It's also horribly expensive to operate, and not remotely the most efficient "all-electric" heat, which is a cold-climate heat pump(CCHP) that can be 250-300% efficient. Also known as mini-splits.

The only "advantage" electric baseboard has is that it's cheap to install. It can also cause fires if furniture or drapes are pushed too close to it.

Many utilities have rebate programs for installing a CCHP. Some states do as well.

Depending on utility rates, natural gas or CCHP are the most cost effective form of heat to operate in most places that have natural gas available. As far as I can discern from your question the area of concern for that would appear to be the domain of the gas utility, so perhaps they would be willing to investigate if you are considering connecting to their service.