I've done this in the past, but my reasoning and execution was different. There wasn't a lot on the market and what was there was going quickly. I wanted to make sure that I didn't have some period of time where my kids were essentially homeless. A good mortgage person can give you solutions to minimize the hassle of switching and having different loans on different homes. What we ended up doing was prepping our house to be on the market while we were looking. As soon as we found the house and had an accepted offer, we put our current home on. Luckily, everything went as planned. I ended up with both homes for a couple weeks and was able to casually move everything over.
A house that isn't being lived can also signal that someone else has moved on and needs to sell. So, be careful how not lived in it looks. You really just want it clean and with good furnishings.
The downside is that it will cost you loan origination fees which are probably $2,500 or more. If you could buy the new house with minimal down, some loans are as little as 3%, then you could wait until you sell, dump the cash into the new place and have them recalculate the amortization schedule. Assuming you do something like that it is just one new loan.
Find out how long properties like yours, or at least properties in that general area, are sitting on the market and plan accordingly. One thing is that interest rates are moving around. The 10 year, which is what loans are generally based off of, is currently moving down, but the fed has been increasing rates regularly. If the rates start moving higher, you are going to end up on the lower end, and any property you buy could be a little cheaper. If they continue downward it may help you on your current house and not so much on your purchase since you are essentially a cash buyer.
One question I would have, why the move? With the price ranges you referenced, that would largely be lateral I would think. It will cost you 6% on your sale in realtor fees plus all of the moving and loan costs. If there is something you want out of your house could you not dump the $30k that you are going to lose into it and save yourself the headache?