Our future landlord wants to change the move in date on our lease that has already been signed, and notarized.

You have an enforceable lease, so you shouldn't have to move your move in date back to accommodate the current tenants, who have PLENTY of time to find a new place. Have you asked your current landlord to let you extend your lease for 5 days until June 5 (at your new landlord's expense)? If so, you have no real problem. They may even require an entire month's rent, but if so, your new landlord should bear that cost too. If your current landlord says no and you have to move by May 31, tell your landlord you can't agree to amend the lease. You run the risk of starting things out badly with the new landlord, but you're only other alternative is to move twice and require the new landlord to pay for the damages you'd suffer, which means a hotel, movers, storage, and a kennel.

While your new lease should definitely be valid if it is as you have represented and your landlord is obligated to follow its terms, your remedy for the landlord's breaking the agreement by putting it off a month would be the actual and reasonable costs you incur for the difference in the date your new place would be ready for occupancy. The most important factor here would be for you to look at the new lease terms. Is there anything in the lease itself that covers what happens when one of the parties doesn't meet its obligations? I'm sure there are provisions for what happens if you wouldn't pay on time, but is there anything in the document itself that covers what happens if the landlord doesn't keep the terms? For instance, is there anything in the lease that would require the landlord to rebate a portion of the rent if the apartment isn't fit for occupancy? If so you should be able to rely on that to get the landlord to pay you the proportionate amount for the time you don't have possession (and that should run from May 5 forward). In addition to asking your current landlord if he might extend your lease by five days, have you considered asking the new landlord whether he would consider changing the date to start the new lease sometime before the end of May, to allow you to move before your current lease expires rather than his proposal of starting exactly one month later? If so, you should both execute an amendment that specifies the new start date. If you don't want to do what the landlord wants, there's nothing that would force you to consent. With any landlord-tenant relationship, however, I'd counsel you to try to reach an amicable agreement that will allow you to move with the least amount of inconvenience to you rather than go in with demands from the get-go. Remember that you're just starting the relationship with the person who's going to be responsible for maintaining the building in which you will live. Trying to reach a resolution that will not completely alienate the landlord from the start would be in your best interest.