Thank you very much guys. I appreciate your help.

I've managed to fix the problem. The first dual capstan deck was Sony
TC-K690 and the second one, the one I still have, is Sony TC-K750ES.

In both decks I've cleaned rollers and heads immediately after first
turning them on. I've used salicyl alcohol and plastic sticks
encovered with cotton wool. After that I had started using the decks
and found problems with tapes curling I have mentioned before.

Just a few days ago I have noticed that supply pinch roller of the TC-
K750ES I still have has white spots on the outer side, so I have
cleaned that roller once again. As I wasn't able to clean it using
salicyl alcohol alone I have carefully used nail-file.

After that I have tested how the deck's mechanism behaves on the
problematic tapes and found out that the problem disappeared.

I don't know what formed the white spots; maybe they were formed by
salicylates(?) I have cleaned the roller one more time, but this time
using isopropanol to ensure there won't be any further surprises.

I think it would be a good idea to replace both pinch rollers as their
surface has microscopic cracks.

Anyhow - I am glad it works now.