Hi roker
What stage is your buyer at?
If they have just put their house on the market, then you should not have accepted their offer.
If they have just gone sale agreed without an exchange of contracts, then you should not have accepted their offer.
Of course, if you had no other offers, then you lose nothing by accepting their offer.
In any event, by all means put in an offer on another house. It's unlikely that the sellers would accept your offer. If they do, you can put down a deposit in perfect safety. But your solicitor will not allow you to sign the contract until she is fairly sure that you will have sold your house and received the proceeds by then.
If their solicitor has confirmed that they have exchanged contracts with their buyer and therefore the sale is legally binding on the purchaser of their house, you are in a better position. Then the seller of the house you want to buy should be able to accept your offer and you should be allowed to sign the contract by your solicitor.
Brendan