Is drifting analog common problem of PSP? How to fix it?

I have bought a secondhand PSP (E1004 Street) a few weeks ago. Generally, this handheld works well, but after a few days of casual playing (max 2h per day) I see a little problem – the analog is drifting, mainly to the right, so in some action/platform games it could be deadly – controlled character can fall down from various places.

I had another PSP some years ago and I clearly remember that I had the same problem then. It was PSP Fat 1004 model. I broke it completely when I tried to fix this issue. So I have a question – is it a common problem that analog drifting to any side? How to avoid that? Or maybe I have a lot of bad luck and I just bought a problematic PSP twice?

I also tried to calibrate it by entering to System settings->System Info and rotating analog, but I read somewhere in the web that then should appear the rotating circle, the same as appears when something is loading, but I did not saw it.
I read also about replacing some rubber from under analog to the new one – does anybody did it? Did it help?

EDit: It seems, that in some games it occurs very rarely, in some other it has never happened, but in another, it makes impossible to navigate even in the main menu – caret jumping through all of the options (in practical game characters almost always move in a random direction without touching analog by me).