By the way, the funds will be disbursed via a method designated by you, the seller, on the listing contract. Your options include a cashier's check, a trust account check from the closing company or a wire transfer to your account. That sum will be for your remaining equity in the home, of course, after all expenses and fees have been deducted.
As for the pending purchase of your next home, you can try adding a contingency to your purchase contract saying you won't close on the deal until you first sell your existing home. That tactic is more acceptable to sellers in a depressed home-sales market with few competitive bids than in a hot market. But if there's pressure from other bidders and you must act quickly to avoid losing the house to another buyer, there are ways to come up with a down payment without the funds from your sale.
But I must add that it's much safer to first secure the sale of your existing home because it's always the biggest "if" in the sell-buy scenario. The worst that could happen there is you'd be temporarily "homeless," taking up residence in a hotel, apartment or at a relative's home until you complete the purchase of the replacement home. Buying the new home first, on the other hand, could put you on the spot for paying dual mortgages much longer than you originally bargained for, particularly if your "sure thing" buyers suddenly can't qualify for a mortgage, or they find a way to break your contract to pursue a better deal -- scenarios that are all too common.
But if you truly need to buy the new home first, you could request a "rent-back" from your buyer. That's where you remain in the house temporarily and pay a monthly rent to the buyers that's roughly equivalent to their new mortgage payment, plus insurance, interest and taxes. That way you pack and move only once. Failing that, you could try to arrange a bridge loan, also called a "swing loan," to use as the down payment on your replacement home. However, these loans tend to carry strict qualification guidelines and can be much more costly than your standard equity loan, plus they're tough to get these days. Another solution is getting a home equity line of credit, or HELOC, to make your down payment. These typically run a point or two cheaper than bridge loans and don't require payment of any closing fees.
Good luck! Here's hoping your dual transaction comes off flawlessly.