My own view on razing is that I will only keep a city if
- it has Wonders that really help the warmongering (Machu Pichu, Pyramids, Notre Dame, Chichen Itza, etc)
- it has a good population and science buildings, so I can annex it (and have the happiness to do it, which you don't)
- it's strategically necessary to puppet it in order to assault another city
- - -
But beyond those questions, I'd encourage you to look at certain things that your screenshot reveals:
1. Troops. You don't have very many. This will slow you down. The faster you can conquer, the easier it is, believe me.
2. Science. For T300+ your science is really small. I don't know what difficulty that is, but on higher difficulties you need to reach certain military techs in good enough time for them to be properly effective, and to stop the AI from winning. Look at some of the threads on this board to learn how to grow your science better.
3. Happiness. If you take Autocracy and Commerce (which you should for every Domination sweep), you should not be unhappy. Unhappiness actually applies a penalty to your units combat strength and they could lose the war as a result. Plus, barbs could spawn next to your undefended capital and pillage your stuff.
If I could recommend one video that includes late game conquest, I'd probably choose PeddroElm's video of the Songhai Deity Challenge
https://www.youtube.com/watch?v=JvN3hIa_hi8