Do transaction fees remain stable in bitcoins, or adapt to purchasing power?
Your understanding is correct. If the required fee rate on the network remains the same (denominated in BTC) and the value of BTC rises against USD, then the transaction fees rise in USD terms.

In reality, there are many factors which affect what the actual required transaction fee will be, such as time/day of transaction (there's generally less competition for block space over weekends), size of transaction and if the transaction uses segwit (since segwit witness data is discounted).