Registering High Engine Temp/Car Heating Issues

My 2005 Jeep Liberty has been sucking my wallet dry since I've had it. It has 136,000 miles, so I'm glad it's lasted so far. Last summer it blew a head gasket and we replaced it along with the thermostat etc. Now that it's cold out, I noticed when I turned on the heat that it was taking longer than usual for the air to turn hot, and that instead of gradually getting warmer, it just switches from blowing frigid air to hot air about 20 minutes into driving with the heat on. Additionally, the two buttons where you select cooling or heat, the heat button shuts itself off after the heat finally kicks in and the air goes cold again and the process starts over. This was merely an annoyance until I noticed the thermostat was registering high again. Even with the heat blasting as I drive, the needle fluctuates between midline and H, and the heat may or may not come back on after shutting itself off. Considering the thermostat, head gasket, and related components (I'm not a car expert) are literally brand new, I have no idea what the issue could be.