Could it help, when I would buy a sd card and put it in it? I don’t think so, but should I try?
Can I wipe the cache in the recovery mode?
I don’t know whether this would help in this situation, I think the OS needs a user interaction to format or at least acknowledge it first.
And incorporating an SD card into Internal Storage is strongly discouraged these days anyway, see sdcardguide, while using it as external/portable storage would not help a full phone storage.
I don’t know the current capabilities of the stock recovery, I’m using a different OS and recovery.
But if cache is still a partition of itself (I only have old knowledge, I’m afraid) you could try to wipe it in fastboot mode with fastboot erase cache.