Couple of suggestions:
First, I would ask the new employer if they provide moving expenses depending on the answer you can better budget how you want to move your stuff.(like moving it yourself or hiring a moving companY).
Check in google, there are many companies that will give you free quote to move stuff. example www.allied.com.
You can use pods (storage box), they deliver to your house, you fill it and they pick it and ship it. Try upack.com
I am assuming that you are not contemplating driving a U-haul, but if you do I would rent the truck from Penske or Budget NOT U-haul.
If you plan to move back to NY in 1-2 years then you also rent a storage in NY, and leave it there for that period of time, but just taking the essentials.
To move cars you can use
https://www.autotransportdirect.com/, they were pretty reasonable.
I moved from WI to GA, but I drove a Penske truck and used autotransportdirect to move my cars(2).
Good luck.