New iPad to replace old iPad that will not turn on and inaccessible?

What you need to do and what you should do are different things. ;-)

You don't absolutely have to do either of those. It's just that if you don't the iPad might still have some of your information on it (albeit hard to get access to), and would be unusable to another person. If you are interested in selling or giving away your iPad then there are steps which you should do so it can be used by another person. However, doing these on the old iPad is not going to affect you signing in on the new device. Frankly I have 5 devices signed into my ID and each time I wanted to add a new one I just signed in on it and left the old ones as they were in case I wanted to use them.

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

What services do you feel you need to turn off? I can't think of any that you can't just simply use on a different device by signing into that new device with your Apple ID. Many services are set on an ID, not actually "on" a device itself.