Not speaking from a professional background in this area, but ...
Nothing will secure the property short of the 10% deposit and a signed contract by BOTH the purchaser and vendor. You can talk to the seller all you want about increased deposits, but without a mutually signed contract, do not consider the house secured.
To make that happen financially, you need you 10% and you need your mortgage loan offer. Your circumstances may mean you need bridging for the 10% and maybe even bridging for whatever equity you have tied up if you want to close on the new one before closing on the old one. However, remember that your loan offer itself may have been made on the basis of a contract signed for the sale of your old house. If that is the case, you are stuck. I have found situations like that where the bank is comfortable with the new situation but will not bridge for the interim period, in which case your purchase is at the mercy of the sale of your own.