My husband and I have just done the same thing that you are considering.
I had no issues selling house in my maiden name but using my married name to buy the new house. None of my banks or ID docs were in my married name as we only started the process a month after getting married. I gave all docs in my maiden name and sent across my marriage certificate and that was enough to proceed with my new surname.
Re equity - apart from a survey fee and a small amount of up-front fee to the solicitor 100% of the deposit & equity I am using for our new house is coming from my sale proceeds less mortgage redemption costs. We have applied for slightly more mortgage than we need so that we get some equity back in cash to cover decoration & new furnishings etc.
To work out what we could afford, as a starting point. I went on a few bank mortgage websites (Barclays, santander etc) and used their ‘what can I borrow’ calculators. Be honest with income & current loans/credit card debits and we found that the answer was very similar to what we got with a broker.
Add the mortgage figure plus your existing equity together and that is a good starting point for your new budget (assuming you are happy to use your savings to cover any stamp duty, legal fees and surveys etc!)
We didn’t need to get a mortgage agreement in principle, but our mortgage broker issued us with a certificate of affordability which basically stated the amount of mortgage we could get so that we could give that to any agents that wanted it.
We did look at properties for sale before ours was listed, but our house was in a very desirable area and was predicted to sell fast. We only had 1 agent that refused to show us a property, all others were fine as we reassured them that we would list as soon as we found a house we were 100% interested in. As it was, we found a house we liked and listed mine next day, but that house then went off the market. We had a proceedavble FTB offer within 5days of listing and the next week found somewhere we also liked.
We only appointed solicitors once our chain was complete, but I did to research on who to appoint before then. The ‘reallymoving’ website came up with good quotes for me, and seemed to have genuine reviews if you are stuck. I went with an ‘online’ solicitor based in Stockport (100s of miles from where we live) and they have been great!
My advice would be to do the math and have a look at properties in your budget. Be upfront with agents and let them know this is what you are doing to see whether it is worthwhile you moving yet.
Best of luck!