How should I upgrade? A new preamp with a phono stage or just a phono preamp?

Would recommend a separate Phono preamp. My experience with integrated Phono pre's is that's where they cut corners. Plus if you change cartridge or TT at some point you don't need to mess with your system's. pre amp. 

Good luck!

The Audible Illusions M3A is a fantastic unit. I owned one some years ago with the MM Phono stage. The lack of a remote control was the deal breaker at the time. The AI L3A/B has a remote yet no phono stage.

Chances are, you will not be disappointed in the M3A or B. For the money the M3A’s are going for, it is a steal!

I have always been rewarded by incorporating the highest quality preamp and separate phonostage I could afford. Used Audio Research and Conrad Johnson tubed preamps and phonostage are among the best made. Going backwards in time and buying the best you can afford will award you with great, natural sound.

There are several ways of looking at this and it all depends on what you like and where you see yourself going with the hobby.  Any good stand alone phono stage from any good company is going to outperform the built in phono stage of any integrated/preamp or receiver.  So if you buy a good stand alone phono stage now it would serve you well going forward no matter what upgrades you might consider in the future.  On the other hand, if you simply don't want the clutter, like the simplicity and elegance of a single piece, and there is nothing wrong with that, then go that way.

I had a vintage nikko nr 719 receiver with a built-in phono stage and a technics turntable and it played marvelously. Used with epicure linear speakers. Probably sounded better than most of the stuff out there in the modern age. Components were really built well back then.