It seems that you've made a few mistakes and the landlord is taking advantage of it.

You should have communicated them in writing with the signatures of all persons that signed your contract, e.g. if you and your wife signed your contract the communication is only valid if you and her sign the letter. This letter should be sent to them by registered mail.

The earliest date that you can leave is canton dependent (I think), in canton of VD is minimum 30 days. Search for the specific case in your canton.

Early return of leased object is according to the Art. 264 of the CO. In order for you to be released from your obligation you need to find a replacement tenant who is solvent and who are willing to take over your contract at the same conditions, and only one solvent tenant is enough. By "the same condition" means the same price, terms and everything else and depending on your contract conditions you can even take this as advantage, for instance if your contract is old, has a higher "reference mortgage rate" than today and you have never requested a rent reduction you can advertise that fact (this is what I am doing BTW).

https://www.droit-bilingue.ch/rs/lex...264-de-fr.html

Make a form gathering the relevant data of the ones willing to take over your contract (I have one in French if you want), ask the ones willing to take over your contract the important documents, usually copy of their ID or Residency Permits, 3 last pay-slips and the "extrait office des poursuites" (sorry don't know the term in German). And send these documents to the landlor by registered mail. You will then in principle be released from any further obligation even if the landlord refuses the candidate you proposed. The only case you still need to find a replacement and continue paying the rent is the candidate is offered the same contract conditions and he gives up. It is worth noticing that if the candidate is proposed a 30% increase and he gives up you are still released from further obligations but in this case, ask the candidate for proofs so you have something in case of problems.