Hugo wrote: but with the 6'0 (it has more width and thickness) I could stand up but the board would go under water

This is because the board isn't moving fast enough - get enough speed and you can surf anything (people can barefoot water ski on nothing but their own feet). On a longboard, there is enough volume (floating you above the water) that if you aren't moving very fast you are still surfing. But on a short board/lower volume, lose speed and you sink. This is why I mentioned the pumping for speed - you just cannot stand on the board and let the wave do everything. Esp on a shorter board and at your weight.
Its not an insurmountable issue but it will absolutely require you to learn new skills which may take some time. And if you are surfing 2-3 foot waves, the risk is that you need to get to a certain skill threshold in order to actually surf those waves, but cant get to that threshold (ie learn how to do it) because you cant actually surf the wave because you don't have the skills in order to practice - a vicious circle. Single fins make this harder because pumping on a single fin is pretty hard (nothing to push against, unlike a thruster with its side fins)
So more volume as Big H says, is your friend. I can't see to many downsides to more volume esp if you are going single fin
Single fin is better than twin fin in my view. A twin fin is pretty much designed to go fast and its not great for turning - a much more limited set up, in my view. Quads offer the same benefits as twins without so many disadvantages, but again its a different style of surfing to singles. Go single, and if you are having speed issues you might be able to put in some knubster side fins (if your board allows for it).
Also, don't change boards because of your stance. Change your stance!