Hi tarnator,
While sheets and bedding and good sources for them are not my area of expertise … I do have a few suggestions that may help.
It seems that in these days of thick mattresses … the shallower pocket sheets that were more “standard” not too long ago are harder to find. Fitted sheets that have gussets that are a few inches more than your mattress should still be fine especially if they have good elastic all the way around the edge of the sheet (not just in the corners). You would probably be fine with a pocket up to about 15" (made to fit a mattress up to about 12").
Dreamfit has good feedback about their sheets and staying in place and have re-inforcing elastic straps in each corner.
https://www.cozytownlinens.com/ Sells a range of sheets with custom pocket sizes (from 6" to 25").
https://www.hardtofindsheets.com/index.htm also has some hard to find sheet sizes and thicknesses.
Sheet straps/suspenders (or garters or clips) such as some of the ones that are listed in a google search for sheet straps can also be a good solution to keep sheets tight on the bed if they are either too big or too small. This may let you use sheets that have deeper pockets. Drawstring sheets such as these may also be a good option.
There is some information about different types of sheets in post #7 here and there is more information and a few good sources for sheets in post #26 here. In many cases sheets that have elastic all the way around will fit and stay in place much better on thinner mattresses than sheets that only have elastic in the corners.
I also did a google search on “shallow pocket sheets” and I was quite surprised that there were as many sources as there were so there seems to be some hope for those who own thinner mattresses 
Hopefully some of the members here with thinner mattresses will have some suggestions about sheets that have worked well for them as well.
Phoenix