They sent you an offer letter. Presumably, you sent a reply, acknowledging and accepting their offer. UNTIL AND UNLESS THEY HAVE TOLD YOU THAT THEY RECEIVED AND ACCEPTED YOUR REPLY, you do NOT have a job.
Many years ago, I saw a case at my employer at the time, where an offer letter was sent, the candidate telephoned to accept, and THAT was when the employer learned for the first time that the candidate was essentially unable to communicate verbally in English. His accent was too thick. Unfortunately, the reality was that fluency in English was an absolute requirement for the job. The offer was rescinded.
Contact them and explain that, because of lease issues and notice requirements, you need them to expedite your contract. Explain that, without an expedited contract, you will not be able to start on the intended date. Explain that this is a difficulty for them, but you are certain they will understand.
WORST CASE, the job offer evaporates. At that point, you will have dodged a bullet.
If you resign and terminate your lease and start your relocation, without a signed, enforceable contract in your hands, you are over the proverbial barrel when they decide to modify the agreement, as Darth Vader put it.