Last minute lodging in Andalucia- late February/early March

Should be no problem.

Fasching/Carnival is February 24-25 in 2020. Hint: School Holidays. Easter is April 12 in 2020. You have some time left to book so take advantage to do some more research. Whether you book a hotel(s) through them or not, using Booking.com to scope out what is available for a locality/set price point/quality is a good way to get started seeing what is available. You can plot out several itineraries and see what likely hotel costs will be for each one. If you pick a hotel on the site go to the "show it on a map," and all other hotels in the area will appear with their prices for comparison.

But you are two months out so you don't have too much extra time given the Northern Europeans who like to escape south late in the winter.

Andalucia is huge, so it depends on where you’re going and how you intend on traveling. If you intend on visiting the main tourist towns you should have no problem. However, many small towns along the coast that are full of tourists in the summer, tend to essentially shut down during the winter months. By that I mean, many shops, restaurants, and small hotels will be closed.

NickB gives useful advice about being careful about the Carnival holiday, many Spanish people will take advantage for a few days away as well. We did an extended road trip in Andalucía last October mostly booking at the last minute (a few days ahead or even a couple of times in the morning for the evening) while we were travelling. We were on a limited budget too and used mostly Airbnb and some Booking.com. The only time we ended up not stuck but staying somewhere not ideal was for the long weekend of 12th October, which we hadn't realised was a holiday. I think it depends on whether your itinerary is flexible or not, too. For example if you've already got tickets for the Alhambra you might as well get accommodation booked too. It can also be a waste of holiday time to have to decide on accommodation while travelling....