I am replacing a RD-TX35. I ordered from Amazon directly and received the RD-TY300 (which I understand is the upgraded version).

My question is about how to mount the RD-TY300. It seems to mount not on the frame but to the wheel mount with the claw.
Also, some versions have a silver "adaptor screw" next to the claw. Why is this?

If it has a claw mount then it is indeed meant to be clamped in place by an axle nut.
The ones with an extra screw are meant to be used on frames with horizontal dropouts. The screw, together with a special washer that sits in the dropout, lets the derailer stay in place even with the wheel removed.

I'd return it and get the correct one for your bike. It sounds like your bike originally had the direct mount version.

Shimano makes the ty300 in three different mounting configurations.