Part of my job is helping and advising people get their homes, (then rebuilding, restoring and maintaining)
In this area, I would certainly advice that you get rid of one house before attempting to get your new home, you would be very unlikely to get an offer accepted, if you have another to off load,
Quite a few of my clients are people moving to the area, and find they need to rent until they can make offers, demand out strips supply, so any seller is only going to entertain offers that come with no complications.
Something that happens here with increasing regularity is lower offer being accepted , which is extremely frustrating for those who are offering more, people do this for a variety of reasons, some social engineering, some because of cooperation.
A lot of properties in this area never go on the open market, people tend to register an interest, then when the owners eventually want to sell (sometimes years later), they contact the interested parties directly, who put in bids.
We have a lot of people waiting for the right property to become available, usually a lot of compromises have to be faced.
Any signs of heel dragging (such as surveys) would eliminate you from the process.