Should I convert from a rooftop to a split HVAC system?
Is this a heat pump with supplemental gas heat?

Sounds like the shuttle valve is stuck or something in the cycle reversing has failed, heatpumps do both AC and heating.

Why double the misery by having two systems to fail and not having the current one repaired/replaced?

As to new systems, you can get very efficient heat pumps with supplemental gas heat, we just installed such a system in our business. During low level heat needs, it runs in heat pump mode, when the temperature outside drops below a certain settable level, the gas portion runs in tandem. The gas section is the new high-efficiency type that extracts so much heat that they use a PVC stack and water drain for the condensate. You'd kill two birds with one stone, having AC and more efficient gas heat.