Wallets

So. I want to hold noncustodial. But I want to be able to use one wallet to trade, buy, and receive. And at least have a good amount of different coins and tokens to choose from. Is there a noncustodial 'exchange'? I thought I had found one in ShapeShift.io but I can't get the wallet to work on my Android phone.

Suggestions, is there anything out there?

Note: As a bonus it would be great if they were to incorporate a 'pay with crypto' feature (like Flexa w/Gemini (I understand Gemini is still custodial)).