17 Days "Last Minute" Trip to Spain

Can you fly into Granada and out of San Sebastian?

Once you choose either Basque region or Andalucia, it will all fall into place--even with 17 days, I would not do both because you have to pass over so many great places to do so. It is a hard choice to be honest, both are pretty nice!
Weather can sometimes decide it, but in May I would weigh them equally (I have little issue with heat in May--based on one Basque trip in May and two in Andalucia in May). I'd play around with open-jaw flight itineraries to first see what maximizes your time and budget.
You could start with Valencia and branch out from there by car, then drop it when you get to Andalucia, where transport is great.
You might need the car more in Basque region, depending on your ultimate choices.

I have been to almost every region of Spain, except Valencia.
I would suggest not trying to do the Basque region, but focus on Valencia, Granada, Rhonda, Seville, Cordoba, Madrid (day trips to Toledo and Segovia).

I rate Toledo and Seville and my favorite cities in Spain, do not miss them.

Do the Basque region when you can also visit the Bordeaux and Carracassone region of southern France as well on another trlp.

For reference here is trip report for our 11 night Spain visit in Sep 2023. https://community.ricksteves.com/travel-forum/trip-reports/trip-report-spain-aug-sep-seville-ronda-granada-cordoba-madrid

It's likely to be getting quite hot in Andalucia by late May--if not by early May, so if you decide to head south, I'd recommend pushing that area into the earliest possible section of your trip. In truth, May is really not the ideal time for Seville or Cordoba. Granada's elevation is somewhat higher, so it is usually a bit cooler.

Seville weather -- May 2024

Weather varies from year to year, of course, so I recommend checking the most recent five years. Use the pull-down box at the right, just above the graph, to change the month and year displayed. Use the Search box at the upper right to change the location.

The Basque Country tends to be cool and overcast (sometimes rainy) even in the middle of the summer, so I'd try to go there toward the end of your trip if you opt to see that part of Spain.

Our trips to Spain focus on smaller geographic areas. It would be easy to spend your entire time in the area southeast of Madrid. That would include places such as Zaragoza, Teruel, Albarracín, Cuenca, and Almería. With a vehicle there are a plethora of small towns and villages worth visiting, as well as places to do some walking/hiking. It just depends on what you want to see and do.

When we planned our Spain trip, I included Valencia after seeing a youtube video and I am so glad we did. We had 14 nights, flee open jaw Barcelona 5, Valencia 4 and Madrid 5. We prefer longer stays and did not want to force visit to Andalucia. You have 3 more nights to juggle.

I would suggest similar route and see if you can fly open jaw into Barcelona (or to Valencia if you decide to skip Barcelona and add Basque region) and home from Seville (or vice versa). Spain is a large country and careful planning is needed yo avoid doubling back over long distances. Much will depend on what flights are available from your home.