Should I upgrade from Sylvania 6DJ8 to better tube

You may get different sound. If you want more detail or a "warmer" sound you can switch. 6DJ8's aren't cheap but run for many many years. They have a great impact on the sound and can be fun to roll. I'd go for Mullards...

Generally Mullards warm, Telefunken neutral, Amperex somewhere in between.

Each brand will 'tailor' the sound somewhat.
Amperex is a nice overall tube, but there are those made in Holland, USA, Japan,etc; they will all have their own characteristics...which makes tube rolling fun.
Some tubes will give a warmer sound, some leaner.

Don't forget the 7308 tubes, which usually provide better highs but not quite as strong bass compared to 6DJ8/6922 tubes.
If you search around you can still find good deals.

In my experience, there is substantial improvement over a 6DJ8 tube by going with a 6922 or 7308 which offer very tight triode sections.

Two top choices are 1960's Siemens 6922 and 1960's US made Amperex 7308 (either orange or white label). The Siemens is very clean, focused and slightly forward and dry, while the Amperex (especially the earlier white label) is holographic, airy, and presents a wonderful midrange. You could hardly go wrong with any of the vintage European tubes...