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Sounds like a problem I had a long time ago. Replace the blower switch to your heater. That cured my dilemmma, in fact with the new switch, the low speed blew heat better than the old switch did at high speed!
The cutting in and out I hated, someone told me that the contacts can get corroded too. However, the work involved in removing the switch to check or clean the switch contacts, I would just rather replace the switch.
I purchased my new heater switch at NAPA, and it was around $11 at the time, if I remember.
Good luck. Might be time to flush your cooling system as well, if you have not done this recently.