I've lined-dried all laundry, year round, for all of my adult life. So my first thought was "Suck it up, Buttercup."
But....
I do have a few pack towels lying around from our backpacking days. They are thin like Turkish towls, yet more absorbent. I don't recall them being more expensive than other types of towels.
We do have ample space in our small home to always have a drying rack set up with things on it, so even if something needed 24 hours or more to dry, it's not an issue. My spouse's socks need the most time. We also leverage pants hangers to hang a bath towel, so it can hang anywhere, while pulled into a rectangular shape. Might be difficult in a rental, but if there is a hook or curtain rod near a sunny window where the towel could hang from a hanger, it'd work.
Love your username.