I've had the same exact issue, only the other way around.
I used to live in the USA and I just moved back to Europe in January.
What I would recommend is
1- move physically
2- open accounts in the USA
3- once you are set up, sell on one side and buy on the other.
If you don't have a lot of extra cash you will probably have a few days in between, but the change should not be that dramatic.
If you don't have any USA account now, do not underestimate the difficulties of opening one because:
You need a proof of address and a social security number to open a bank account in the USA
Nobody is going to rent you an apartment if you do not have a bank account and a social security number
On top of that, USA banks do not "talk" to European banks, so you will most probably start with ZERO credit even if you are a millionaire somewhere else.
My recommendation is to open an account with Hsbc ASAP in your current country, as it is the ONLY bank that has the different subsidiaries communicate among each other, and they will immediately open you an account anywhere and give you credit.
It was a huge bonus for me when I moved to the USA, as I started with a small but reasonable $12.000 credit, whereas some other expatriates (even people much much wealthier than me) had to struggle for a couple years with $500-1000
If you're mustachian not having credit is certainly not an issue but it will become one the moment you need to buy a house, as the interest rate the banks will charge you depends on it.
Also added bonus with Hsbc is that if you have more than $100.000 with them you have free "Global View", which allows you to move funds between your international accounts completely free of charge.
I am not affiliated with Hsbc at all, but I have been an expatriate for the last 13 years of my life and came to appreciate their advantages.
Plus, other than that as a bank they kinda suck
