In this market, I would caution my sellers about accepting a sale conditional upon another sale.
The more money you put down at contract signing, the better your terms will be considered in accepting an offer. I have not yet experienced anything accepted in lieu of a check. With a good enough offer and if your credit and income supports it, perhaps the seller might consider taking care of closing costs.
You might also look into a bridge loan. Not only will this provide you with a "pre-approval not contingent upon the sale of a house," but it will carry you across the gap between purchase & sale. The best bridge loans I have yet to see are offered by Coldwell Banker Mortgage (PHH Mortgage). There is an 800 number you can use to get more information (no obligation). (866) 819-5265