You should be aware, if you are not already, that it is not necessary to have a valid visa to remain in the US. The visa only needs to be valid when you enter the US, whereupon you are given a period of authorized stay by the immigration officer. That date may be written or stamped into your passport, and it will be on your I-94 form, This I-94 form is normally a database record available at the CBP site, but sometimes it is an actual piece of paper, especially if you have extended your stay by application to USCIS.
In any event, as long as you leave the US before the expiration of the authorized period of stay shown on your most recent I-94 form, there's no violation of US immigration law, since you're not actually in the US.
(You have reminded me in a comment that there is another circumstance under which it is necessary for a nonimmigrant to leave the US, which is falling out of status. The details vary depending on the specific status and the reason for falling out of it; sometimes there is a grace period and sometimes there is not.)
One tenant moving out of a jointly leased residence may have some consequences for the validity of the lease, or for its renewal, depending on the jurisdiction. To know for sure, you would need to look into the laws governing residential leases in the relevant jurisdiction. If the lease is in New York City, at least, your fiancé can continue living in the residence and can even bring in a roommate. If your fiancé wants to continue living there after the end of the lease, the landlord can probably refuse to treat it as a lease renewal, instead concluding a new lease. If the residence is a rent-regulated apartment, this can result in a greater increase in the rent than would otherwise be allowed, potentially allowing the landlord to deregulate the apartment. Other states and cities will have different rules that may be less (or more) favorable to tenants.