Beechcraft C23 Flight Instructor reported safely landing after experiencing engine power issues associated with fuel system anomalies.

Narrative

We departed ZZZ1 on an IFR flight plan. The right tank had 19 gallons and 13 gallons in the left tank; I as the CFI double checked with a fuel stick. We left ZZZ1 on the ZZZZZZ.ZZZZZ1 flight plan and ATC vectored us as such. We noticed about 10 miles from City X that the fuel gauge on the left tank was slowly losing fuel; but I determined that this was not accurate as we had the fuel selector on the right tank the whole time. (this fuel gauge has had slight problems in the past) We resumed VFR and cancelled IFR with ZZZ [ARTCC] after seeing the airport. About 6 miles from City X; the engine started sputtering. Knowing we were about 3000 MSL; it was between the highway and proceeding to the airport; After my student onboard started flipping the fuel tanks along with me enrichening the mixture and turning the fuel boost pump on; the engine turned on and off; giving us enough altitude to make it to ZZZ. I also requested priority handling with tower and they gave us priority landing and we landed on runway XX safely.I want to note that as the CFI on board I took controls immediately and my student was helping me via CRM.It was determined later on in the day by a mechanic that pistons beneath the fuel selector valve were busted and leaking fuel from the left tank back into the right tank. It is also worth noting that despite having the right tank selected the entire time; (we did not switch tanks; just to keep it on the fullest tank the entire time) when we landed; the left tank was bone dry and the right tank was entirely full.So in conclusion; the engine was leaking fuel and burning fuel; essentially doubling the fuel burn from the left tank. The fuel selector valve was not functioning properly. I want to thank the staff at the ZZZ tower for helping us out along with the emergency crew on staff.