I guess I’ll try this one just to keep it moving along. I don’t think you are “heavy”, I am exactly your size. There are plenty of people much heavier than 200 paddling SUPs out there and there are plenty of 200 pound guys surfing very small, “unstable” SUPs.
There are a bunch of design features of a board that affect how stable it feels. The most significant are width, volume, thickness, and length. I think the two most important features affecting stability are width and volume. Look for a board that is at least 32 inches wide. For volume, at your weight, look for something that is at least 180 liters, 180 may feel a little tippy at first but you’ll get used to it. My 70 year old mom with two fake knees paddles a 32 in wide, 167L board in flat water. A board that is, say, 34 inches wide and around 200 liters should feel pretty solid in flat water. Length is not super critical; most boards will be between 10 ½ and 12 feet long and all other things being equal you won’t notice much difference.
There is more to the story when talking about chop or waves. The above factors are still in effect in chop but your own skill level, balance, core strength etc are very important in chop or waves. That only comes with time on the water, but in my experience it comes quick. So get out there, don’t be afraid to fall in and have fun.