How binding is the contract?

I am in the process of buying a new home. I made an offer earlier this month, on a house that we liked, and the seller accepted my offer, on condition that I take the house as is, as she knocked down $5000 from her asking price of 210K. I accepted this, and we started the paperwork. When the contract was presented to the seller, they further added clauses, about not paying for any sort of inspections (septic or other), and wanted it stated very clearly that the final price of 205 K was for the property in as is condition. They also changed extended the closing date by 15 days, from what was initially proposed. Well, I agreed, and we set a new closing date, and both parties signed the contract.

Now, the seller wants me to extend the closing date for another month, or let them retain possession of the home, a month past the date set up in the contract. I am unwilling to do so, as my rate lock expires on the day of closing, moreover, I have other priorities myself, with my current home, that I cannot extend closing for another month. When I politely tried to explain this to the seller and their agent, they got a bit rude with me and insinuated that they would not and could not vacate before at least a month past the closing and that I would have to negotiate it out with them. I am not willing to do so, but I am afraid that if I stay firm, that the seller may be able to slip out of the contract somehow, since the seller's agent seems totally concerned only about the sellers interest and not mine. What should I do?