Digital Advice

In your shoes with your budget I think you would be best off with a stand alone streamer with it's own built in DAC. Something like a Bluesound Node would fit right into your budget. Its user interface is both robust and intuitive, and supports a wide variety of streaming sources including Amazon HD. It would also provide you with a nice upgrade path should you choose to add a stand alone DAC down the road. Just my 2 cents. The Wiim products would also be worthy of consideration at a somewhat lower price point. For many of us products like these were the gateway to streaming.

I have a WiiM Ultra streamer/DAC and think it is fantastic for the money, which is well under your budget cap.  And you can buy it on Amazon which offers free returns if it doesn't float your boat.  For me, it falls in the "there is no reason not to try" catagory.

This Bluesound Vault will let you stream and you can also load all your CDs into it so you can have all your music from one source, and I believe it also works with Amazon Music. 

https://www.usaudiomart.com/details/650171585-bluesound-vault-2-streaming-music-player-with-2tb-drive-cd-ripper/