Hi mate
Definitely don't give up imho.
From your vids apart from a sharper / faster pop up your pretty much doing what you can on those really small waves.
My advice would be slowly try to go out in bigger waves (say chest to shoulder high). Even take a body board out or go for a swim just to get used to it. Develop some set techniques for getting caught inside so you can stay calm and not worry even when a bigger set comes through. With your board and some proper commitment / determined pop up you should be able to get into a bigger wave pretty early so hopefully not too steep, trim your line then just enjoy the ride

Once you realise your not going to die you'll get more confidence then shoulder high waves will become normal and you'll wonder why you used to be scared.
Final suggestion from me and a little bit left field is to keep a diary of your surfs and conditions. Quite good fun and great to track progress / build confidence.
Good luck
