Possible to do laundry on 14 Day Rick Steve'stour of Spain?

Or do you really wash things in the sink and they will dry in a couple days?

Yes, this. And I wear tops 2 or 3 times before hand washing and bottoms numerous times.

I bring lots of lightweight tank tops and wear a top over them, which varies by season - a short sleeve top, long sleeve top or a sweater

The longer the trip, the more washes, not the more clothes.

Yes, bring as many clothes as you can that can be sink washed and that dry over night, not a couple of days. Test wash and dry everything before you go, but know that humidity and temperature will affect drying times. My practice on tour is to do the washing as soon as we're allowed in our hotel room - it doesn't take long and maximizes drying time.

It's entirely possible your closet already has things that fit the bill - look for rayon/poly blends and not 100% cotton. There are a multitude of posts in this forum about packing light - have a skim through the packing section https://community.ricksteves.com/travel-forum/packing. Many frequent posters also add information about clothing choices in their trip reports.

Likely Granada (a two night stop) will be the best chance to get laundry done, either by the hotel or at a local laundromat. You can call the RS office to ask about your specific tour.

I've done a dozen tours.
Generally I only sink wash underwear, and look for a laundromat maybe once a week. RS guidebooks usually mention laundry facilities available in towns, or I google possibilities.

There has been a time or 2 when there has been a chance to get laundry done inexpensively by drop-off. The guide would mention this possibility at the start of the tour.

Rather than t-shirts, bring cotton blouses. They dry quickly, like overnight, when washed in the sink. Use your towel to get out extra water and do a quick blow dry of them. Panties and socks, same thing. I do have a plastic bag to put things in if they aren't quite dry and hang them up at the next hotel. Wear jeans and you will never have to wash them in 2 weeks.

Generally the guide will advise on the best places for laundry, and be able to give directions. Sometimes it’s self-service laundry, or can also be a drop off, depends on what’s available. Not always at a hotel though. Once you get your hotel list you can do some research of your own through the hotels’ websites. Or Google maps for laundromats.

If you pack lightweight clothing that takes up minimal space, you may not need to do any laundry.

I haven’t been on a RS tour, but on our longer trips I look up laundromats in towns where I think it will be time to do laundry and bookmark them to Google maps. I try to find one near our lodging. Once you get a hotel list you can look them up. For Spain search on lavandería autoservicio.

Yes, there will be an opportunity to send out your bag of dirty laundry to be washed by the hotel.
If I remember it was around 25 euros for a bag and we paid cash to the hotel.
I know we did it during our RS Best of Spain.
But I just can't remember in what city we did this. It was definitely in the middle after about 7 days.

We’ve been on 10 RS tours. Rarely do we wash things in the sink at the hotel other than an occasional pair of socks or underwear. We go to a laundromat about half way through the tour. We have fond memories of both locals and ex-pats we met at the laundromat. You can do this and fit everything into a carryon. You do not need to take nearly as many clothes as you think you do. I take 4-5 tops and a cardigan. Merino wool is my favorite. I can’t say enough good things about Smartwool t shirts. 3 jeans/pants/skirts depending on your preference and weather. I take one of each. I never take shorts and neither does my husband. My “splurge” when packing is a couple extra pairs of panties and 4 pairs of socks instead of 3. I cut back my list after every trip!!