I also live in an apartment and have to go to a laundromat as well. We have enough clothes that I only do laundry every 3-4 weeks (I HATE going to the laundromat), and I definitely don't have enough liners yet to make it between laundry days.
I rinse mine out really well with warm water, wring them out as best I can, and let them air dry (usually takes 2-3 days to dry out completely). The smell doesn't go away COMPLETELY when you hand-rinse it like this, but in my opinion, it's good enough for one more use until I can use a washer. I change my liners twice a week.
I've been playing around with what I use for the middle layer. I started out with the quilted batting, since I learned to make them using the instructions that Stackie posted. I tried one liner with blanket batting (that comes rolled up in a bag) and that was a massive failure, the pee just trickles right through it to the bottom layer. Right now I just made 3 more liners using cheap ($1.88) towels from Walmart in the middle. I'm using the first one right now, I'm excited to see how it performs.