Month-long game jams do exist, but my impression is that they're uncommon. itch.io may have some scheduled, so I would check there.
But I suggest that if you have a problem with overengineering things, that is all the more reason to do short game jams, not long form ones. They force you not to overplan and not to get too attached to your work, because you are not trying to produce a “game of the year” candidate; you are trying to produce something, anything, that is playable and meets the contest criteria. Console yourself with the fact that it is very rare that a game jam game will be turned into anything more than that and in the unlikely event that it is, you will probably have time to rewrite it.
I spent maybe 2-3 hours at most on the design concept for the last game jam I did. After that, design issues were addressed as I ran into them during actual implementation. I don't think there's any other way to approach it. Most of your time has to be on execution, and particularly on bug fixing. This is true even in the long-form jams. Your competitors will not be spending a week on design, they'll be spending a day at most and then two weeks implementing and two weeks bug fixing, because in all jams (arguably in the actual industry, too) execution and polish are far more important.