I'm very sensitive to textures too, and for me it's specifically solid fat which I can't stand at all... I don't mind oily food strangely enough, but any fat on meat which is not completely crispy also makes me gag. Chewy or wobbly meat, such as gammon or most British sausages I also struggle to eat, so I'm guessing it's because of the high fat content, since I'm fine with slimyy vegetables, such as courgette. I also don't like floury or powdery textures, such as new potatoes, or unmelted cheese. I really like cheese when it's melted but solid cheese I've never been able to eat. On the other hand, I also love certain textures, particularly crispy food. I have a bit of an obsession with everything deep-fried, and most of my favourite food are crispy or chrunchy in some way.
So, you're not alone, and I grew up in a family where there was quite a lot of variety in cooking and my dad also got me involved in making food from a fairly young age. But I've always been a very 'fussy' eater and there are lots of things I don't eat, due to texture, or to the flavour (many flavours are just too intense for me). I've also been told I've got a really fine palate, which comes in handy when cooking as I can always tell exactly what flavours are missing, and I'm pretty sure all of that is due to hypersensitivity. I can't wait to tell my family that there are others like me my pickiness is EDS-related and not due me just being difficult