Shall we go to the mountains, or we should/should we stay at home?

Can you tell me what the difference between those two is and whether both are grammatically correct? I hear the second more frequently; however, I think both are correct.

a. Shall we go to the mountains, or we should stay at home?

b. Shall we go to the mountains, or should we stay at home?

Hello there, and welcome to the forum.
Sentence "a" is ungrammatical.

Both are unnatural. We would normally use "shall" in both halves.

Shall we go to the mountains or shall we stay at home?

In reality, that would be shortened to:

Shall we go the mountains or stay at home?

To use "should", there would need to be some sense of requirement. For example:

Shall we go shopping or should we stay at home in case your mum pops round?

Those are unlikely options. First, the person would say something like, "Do you want to go somewhere?" Then after getting an affirmative response they would say something like, "Do you want to go to the mountains or the beach?"

Apparently, the word "shall" is used much more often in British English than it is in American English (where it is hardly used at all).