I had the same situation with my apartment, in that it had only hookups. The leasing office told us our choices were to rent a washer/dryer for $35/month, use the coin laundry by the leasing office, go to another laundromat, or buy our own.
I bought a washing machine brand new. It was a basic, no frills one that only cost a few hundred dollars. The delivery was free. Yes, they even brought it up to my 3rd floor apartment. I did not buy a dryer, and instead opted to hang dry my clothes on some IKEA clothes drying racks. It didn't even need a year to break even with the same costs I would pay for renting or going to a coin laundry.
My particular apartment has flat rate water, so those costs would remain the same no matter what. Obviously I have a small electric increase, for both the washing machine and the electric fan I point at the clothes when they're on the drying rack to help them dry better/faster (I dry them indoors since I only have a small balcony and am not allowed to dry my clothes out there per my lease), but I only wash laundry 1-2 times a week.
I will probably be moving in the next few months, but it'll be to another temporary rental. I've noticed so many rentals in Georgia have hookups but no washer/dryer, and this includes where I'll be moving to. This will make it even more worth the cost of the washer!
I could've saved some money by buying used, but it was a great price on sale and I needed one immediately so didn't feel like having to deal with Craigslist.