@Sandu,
I hear you. I too have a very sensitive skin and for a long time was at a loss. Here is what you I did and I am well now;
Buy your own shaving machine and even smoother and carry them with you to the barber
If you happen not to have them,have the guy disinfect his machines while you watch. The shaver is the dirtiest machine I have seen. Ask the guy to open the shaving blade compartment. You will see even shaving powder ya last week. Make him clean and disinfect while you watch( By the way I I get my shaves from a high end barber and so uchafu iko kila mahali)
After the shave,let the guy use a hot towel over the beard area ( you may include dettol etc)
Finally apply a liquid called BUMP PATROL. It is available in supermarkets. Usiweke aftershave for the first few shaves just incase they may be causing irritation. The Bump Patrol is good anyway.
Let me know if this works
I Think Therefore I Am