Re: going through coolant
A/C issue and coolant loss unrelated.
Coolant loss is either external leak (hoses, radiator, reservoir, etc..) or internal leak..breached head-gasket: coolant leaking into cylinders and being expelled through exhaust.However, white smoke not always present.
Do a cold and hot (operating temp) pressure test (rent free tester at auto parts store). Do not exceed spec psi of coolant system.
Do a Block Test (free rental) to rule out head gasket breach.
Could be as simple as a the thermostat is stuck closed or reservoir has developed a leak. Other suspects include failing or failed water pump (can leak when completely failed). Check to see if coolant fan engages.