If I understand correctly, a watch is muktza machmas melachto l’issur. Therefore, it can be moved for tzorech makomo and gufo. Therefore, if it is ok to wear it because it’s tzorech gufo – is that correct?
My watch strap broke and I am no longer able to wear it on my wrist. However, I still carry it with in my pocket and use it. Is it still permissible to wear or is it a stricter category of muktzah that cannot be moved at all?
A regular working watch is considered kli shmilachto l’heter, since the main use of a watch is to tell time, which is a permitted action. Therefore, the watch may be moved for any reason. If the strap broke before Shabbos, you can still carry it in your pocket, but doing so is risky, because you might walk outside where there is no eruv and forget to take it out of your pocket. If it broke on Shabbos itself, then it might be muktza and cannot be moved. This is since we might come to fix it on Shabbos, therefore it is muktza, and may not be moved. However, from the wording of your question it seems that it broke before Shabbos, therefore it is permitted to move.
Rema 308-16, M:B 308-168.