Hi Luke!
It's more likely that the new joysticks are broken too. But, it's also possible that a trace on the motherboard is damaged and causing this.
First, please inspect the motherboard to check if all the traces are OK (especially the traces near the joysticks, where you have soldered/desoldered).
If those are all OK (when in doubt, please share some pictures with us). Then buy some new Joysticks, and when you are there, why not buy the Hall Effect joysticks?? Those are great and take care for good with the drift (drift is no longer possible with the hall effect ones).
You can buy them here: https://www.aliexpress.us/item/3256807143069005.html?src=google&pdp_npi=4%40dis%21EUR%2117.49%2117.49%21%21%21%21%21%40%2112000040289311081%21ppc%21%21%21&src=google&albch=shopping&acnt=272-267-0231&isdl=y&slnk=&plac=&mtctp=&albbt=Google_7_shopping&aff_platform=google&aff_short_key=UneMJZVf&gclsrc=aw.ds&albagn=888888&ds_e_adid=&ds_e_matchtype=&ds_e_device=c&ds_e_network=x&ds_e_product_group_id=&ds_e_product_id=nl1005007329383757&ds_e_product_merchant_id=700878061&ds_e_product_country=NL&ds_e_product_language=nl&ds_e_product_channel=online&ds_e_product_store_id=&ds_url_v=2&albcp=20701318733&albag=&isSmbAutoCall=false&needSmbHouyi=false&gad_source=1&gclid=EAIaIQobChMIp7CB1MuhiQMVAYuDBx30QDmtEAQYASABEgIV1vD_BwE&gatewayAdapt=glo2usa4itemAdapt
Please let us know how it goes!