Not only do they make soft-sided carriers, they now make them on wheels, so you never have to carry them, just guide with the retractable handle. I got mine on Amazon, but I’m sure Chewy, etc. has them too.
I line it with a pee pad and stick an extra one in the pocket and haven’t had any leakage.
I like it better than the hard-sided carriers because the cat doesn’t shift around as much, plus I’m always paranoid about the front gate falling off mid-transit.