This can put some sellers in a tricky situation, but you do have a couple of options.
You can ask your real estate agent to add a contingency in the contract that stipulates the sale of the current property will depend on the acquisition of a new property. Keep in mind that this will potentially keep the house in the market for longer since only a buyer with a flexible closing date will be willing to wait. This could also mean an offer below asking price.
You could try to arrange temporarily staying with friends or family and keeping your furniture in a storage unit while the transactions are taking place. If done right, you could potentially sell faster and close on a better financial deal.