Occult research mystery card (azathoth) states you must spend 1 clue token from the encounter to place it on the card towards solving the mystery. Does that mean if you want to place another clue token you can only do so after gaining another from a different research encountet?

Yes, that seems correct. Each time you solve a Research Encounter, you can choose to put on clue on the mystery card instead of putting it in your inventory. Look at it like diverting the clue token from going into your inventory and making it move onto the mystery card instead.

Yes, you are limited to only placing 1 clue token on Azathoth's mysteries at a time per player.