The second thing I would do is file for eviction. The eviction process is so long the earlier you start, the better.
Your agent need to address this quickly.
In most cases the purchase receipt and escrow instructions stipulate when occupancy is to occur. I suggest, if your Realtor isunable to help you, that you talk with the Realtor's broker.
Assuming that both parties had professional representation it seems very odd to me that if the sellers were not going to be able to honor the terms of the purchase agreement that they would have given you warning much farther in advance.
Something is fishy around here.
You probably need to go back to your realtor and have them explain.
It almost sounds like the seller of the home you purchased needed to be out on closing day if the agreement for temp lease is void. If they are not out, you could have them evicted, but this can take up to 20 days.
I would have your realtor explain to their realtor that perhaps there is no lease and you need in as agreed.
If compromise cannot be had, perhaps they can allow you a room, or the garage, or something like that to store your items or you can arrange that at your old house.
I guess a third, but not as nice option is to hire a storage unit and move twice, once from your home, once to the new home. Stay in an extended stay hotel until your seller moves out.
I guess though if you start moving your stuff to their house and start living there, they will get out quickly. They typically don't want to share. Have any idea where they are going?