I think you need to aim for something 9ft+, 23 inches wide or wider and more than 3 inches thick. You can
get softtops that are this big, but you can also get 'proper' boards. Two that spring to mind are the
Walden mega magic (fairly expensive) and the BIC supermagnum. But there will be plenty of others - those two are just ones that are 'notoriously' large.
Width and especially thickness are probably more important than going longer - a 9ft board thats 3 inches thick will float you much more than a 10ft board that is 2.5 inches thick. So aim for 9ft - 9ft'6 and then focus on thickness and width, rather than going longer. You arent super tall and if you start getting a 10 or 12ft board you are moving into SUP territory and will start creating other issues
Guys who are 180lb can surf 5ft5 boards which sink up to their chests when sat on - boards that sink when you sit on them are perfectly fine and normal (once you have some experience).
I suspect the sinking when popping up issue is that you are catching white water (which is perfectly fine) and pushing yourself up from the board entirely, whereas on unbroken waves the board drops down the wave as you pop up ie you arent pushing up your entire weight, because the board is falling away. That doesnt help you right now - you need something big to start off - but to give you some 'peace of mind' that once you start catching proper unbroken waves, you can get away with a smaller board. If you want to - there are major benefits to a bigger board.
Another factor might be that you are putting way too much weight on your hands as you pop up and your hands are too far forward - so you are pushing the nose under. It should be a short explosive movement with your hands at chest or lower level. Even at 280lbs, if you put your hands on the middle of a 9ft board and push down, you probably wont be able to sink the board. But put your hands closer to the nose and its pretty easy.
Finally, the faster you go on the wave the more lift the board creates and the smaller the board can be to support a given weight. If you are going slowly (in white water, for example), then the board isnt creating any lift and will sink. So, again - you need a big board but, once you are catching waves, you can (if you want) move to a smaller board. I doubt you will ever be surfing a 6ft high performance board, but a 7ft something will be viable. At some stage in the future. Like in 2-3 years!