OP, this slow rolling of pertinent info and refusal to listen to the pitfalls of your plan are really annoying. You indicated that your problem is that you want to free up cash for living expenses and you also don't like your current home. Posters are trying to help you solve your stated problems.
You also have a misconception that buying an $850K house is going to solve your problems. It won't give you the "few hundred thousand dollars" that you're hoping to net once you pay taxes, realtor commissions, and moving costs.
Plus you'll have to put some amount of money into any new home for storage, some new furnishings because the old ones don't work, window treatments, etc.
A hard money loan is a bad idea for the reasons stated by others.
So how do people sell then buy when they don't have the finances to buy before selling (which is a big part of your stated problem)? They either have a long closing and buy what's available during that time, do a rent back (harder to get with the current market) to give an extra month or two to buy a house, or they sell then do a short term rental to allow more time to look for a house.
You should look at $650K-700K homes in your target area and see if any of them could work for you. If homes in that price range could work, then put your home on the market and either request a long closing or do a short-term rental.
You're lucky with the timing. Your kids could finish the current year at their school, then you could rent a place for a month or two while you house hunt. Then you move into the new house before school starts.