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very worn down...not very protective?
Very worn = not much rubber = lots of punctures.
Gatorskins are (in my opinion) a good mix of hardwearing (even the bog standard version, the 'ultra' or whatever they call it now are even better) and lightweight. Opinions on grip seem to vary massively.
I've just removed an Armadillo as it was starting to wear through the top layer of rubber and seemed to be coming off in chunks somewhat - that's due to skidding though.
Schwalbe Marathon Plus is what you want if you wish to never have a puncture, not what you want if you wish to enjoy riding your bike though, very heavy.
TLDR, a new tyre will be an improvement. You'd need to give us some more info about what/how/where you ride for a more specific recommendation.
Just built a bike and was using old Schwable Blizzard tyres, on my first time using the bike i went to Hyde Park and when i just got there psst, first puncture. This made for a long walk home, then today i went to Richmond Park and went over a pot hole, back tyre went, then when i was looking for a bike shop for a spanner the front goes! It would seem i'm cursed! Disastrous!
So I'm going need some help and guidance buying some new tyres, I've looked on websites but they have been inconclusive. I see Gatorskin Tyres are recommended but i dont understand how protective they are. Do i want kevlar belt or wire bead for best protection? Is it possible that I was just being stupid or are very worn down Schwable Blizzard tyres not very protective? Specialized Armadillo tyres are recommended by Evans. (Bike 700c x 23c)
If you have any new/good condition tyres hit me up!