"Often we go to the museum" - is incorrect! I feel that in conversation the "we" & "often" can be switched... it sounds completely natural to say it this way.
It's possible, but for me it only really works after something else - "On Saturdays we go to the park, and often we go to the museum there". But as a standalone sentence it sounds a bit strange to me.
I agree that it's not incorrect, but "often we go to the museum," sounds a little unnaturally in my opinion. Maybe it needs a comma if used as an opening clause. "Often, we go to the museum." Even then, it sounds like an incomplete thought, like a reason or circumstance must follow why or when we go to the museum. It's just how it sounds to me personally
I completely agree with you @ChlebBoi and I think the other responders are a little misguided.