You're correct that your palate is very narrow - which also means that it's too vaulted.
I recommend fixing it with your tongue, as myself and many others here have done. It pays massive dividends to have figured out how to expand with your tongue because you can keep doing it for free until you fix all your issues.
If you're looking to get your palate expanded professionally, I'd recommend the ALF above anything we know on the market yet. Biobloc, DNA, and controlled arch are also worth looking into.
You need transverse expansion for sure (which is the easiest type, especially with the tongue), but we can't say from these pictures if you need sagittal or not.