The only subject I've ever had to remember quotes for is English, so not sure if I can provide the best advice. However, I remember quotes by:
1) Making a 'TEE' Table - [Technique (if applicable), Example, Effect] and reading over this out loud a few times
2) Writing practice essays/responses while looking at my notes
3) Making flashcards (digitally bc I always lose things) with the prompt of a quote (like first or key word) and effect - You mentioned that you only remember them in chronological order. I shuffle mine or arrange them thematically. Also say my answers out loud before checking.
4) Writing practice essays/responses without looking at my notes
5) Make a note of which ones I messed up or completely forgot, and making a conscious effort to revise these more
6) Where I can, try to make an mnemonic, but when you have like 10+ quotes this can be hard, so only do this where you think it is applicable
* Some studies suggest that reading out loud helps to memorise things. Definitely notice a big difference in my memory when reading aloud compared to in my head.