Things that there should be in Arizona: commercially-made bras that are lined with terrycloth. Things I have not found in Arizona: the identical item.
I am making do with washcloths lining my cups, and I find it much more comfortable than the alternative. It is somewhat warmer, and somewhat bulkier, than wearing the bra without lining, but it pays off massively in comfort, stain-free shirts, and a huge reduction in the chafing from damp synthetics.