My local supermarket is stopping all carriers in a couple of weeks and your only option is to buy their long lasting bags or bring your own.
I always reuse my carriers, long life, short, whatever, till they are only suitable for bin lining.
When I was told the free bags were ending I left most of those I'd brought with me unused and took a few free ones while I can. Did I feel quilty? Yep. But not for long.
In regard to paper carriers; I have read they are more costly to produce than plastic which is why you dont see many around.