1. Set a date by which time if your house is not sold, you will agree to go forward with a bridge loan.
2. Allow the seller to continue marketing their house; if they get a like offer or better, ask them to give you # days to perform or you will let the contract go to the next buyer. At that time, you can active a bridge loan and move forward or give the house up. The seller will not have lost any opportunities.
3. See if you can qualify for a mortgage without the sale of your home.
4. If the seller does not have any time restraints, your offer and terms are good and you are providing a lender pre-commitment with the only contingencies being contract and appraisal, they may agree to wait.
The quick answer to your questions is, yes you can. In previous time sellers usually did not like this kind of offer but as we all know thing have changed quite a bit. One other suggestion if you were my buyer. Make sure on your offer also that to be also contingent upon you getting the loan, if you are getting one of course. The reason for that is to add another layer of protection for you in case a loan does not go through during the contingency period since the lending rules keep changing on a daily basis.