"What is the best/safest method of shipping this many albums to my new home?"
If the records are valuable, drive them yourself regardless of cost. You will never be able to prove what your records are worth if something happens. Even if you declare a very high value, you still have to prove it or you won't get the money.
If you absolutely must ship them, you can talk to your insurance company and see if you can get some type of endorsement added to your HO policy for the move. But its still risky because insurance companies don't like to pay, even when the claims are legit. Whatever you, don't lie about anything. If you do, that will be their excuse not to pay.
Don't use the same shipping company for everything. Send 1 box UPS, another box FedEx, etc...
If at all possible, ignore what I just said and drive them yourself.