Lesser known sites to visit in Barcelona

https://santpaubarcelona.org/en/

https://casavicens.org/

https://inici.palauguell.cat/en/

https://www.museuegipci.com/en/

https://www.barcelona.cat/museufredericmares/eng/museum

Casa Vicens as mentioned above offers an interesting sunrise visit option. Visitors may arrive at 8am sharp (no lateness allowed apparently) then they have 90 mins free reign of this lovely, tiled house before the general public is let in at 9:30am. Seniors discount also honored there. Must reserve for this ahead of time.
Good luck.
I am done. The end.

https://www.mhcat.cat/
https://www.mmb.cat/en/visit/discover/shipyards-barcelona/
https://cosmocaixa.org/ca/cosmocaixa-barcelona
https://www.fmirobcn.org/en/
https://www.museuolimpicbcn.cat/en/
http://www.santamariadelmarbarcelona.org/
https://www.barcelona.cat/museuhistoria/en/heritages/els-espais-del-muhba/muhba-temple-daugust
https://www.barcelona.cat/en/what-to-do-in-bcn/parks-and-gardens/parc-del-laberint-dhorta-92086011952 ---albeit won't be as "luscious" as normally is, since we're in a drought episode and there's less watering in parks.
https://www.monestirpedralbes.barcelona/
https://bcnshop.barcelonaturisme.com/shopv3/en/product/24131/real-academia-de-medicina.html
https://bpa.es/en/

Throw in some suggestions for Spanish shopping

Note that you're visiting Catalonia, the culture here greatly differs from Castillian and Andalucian. It is not quite what you might have in mind -that is, the one often portrayed in "movies about Spain". This most likely translates into your expectation about "Spanish shopping". Can you be more precise? what exactly are you looking for?

Sant Pau is massive and lightly visited but it’s very cool. We stayed around the block from Casa Vicens so we visited early but they only had a line outside on the street in the early afternoon. That area around the Fontana Metro station is a nice choice.

Why not just go to Girona, Figueres, Sitges, etc?

Sunday morning visit to City Hall.
Skip the very touristy Boqueria. The Santa Caterina market is where locals shop. It pretty much closes down after lunch.
The Hash Museum is interesting, traces centuries of hemp use, and has some lovely moderisme decor. Sorry, no samples.
Great churros at Xurreria Laietana straight from the cooking vat to you. Check the hours before you go.
Underground ancient Roman city ruins at the City History Museum
Rooftop tour at the Basilica of Santa Maria del Mar. Interesting for history and architecture, beautiful views, includes entry to the church after the tour.

I don’t think anyone has mentioned the Montjuic Castle (maybe it was in one of the threads but I didn’t read them all). That an impressive site, probably more so because we had taken the Spanish Civil War tour the day before and were more aware of the castle’s significance during that period. There are an informative displays throughout the interior and then amazing views of the city.

We didn’t get out as far as Tibidabo, but if we make it back it will be on the list.

Oh, and if you pass by the Lego store on Passig de Gracie, pop in for a moment to see their replicas of La Sagrada Familia and the facades of other Gaudi sites.

If you visit the Montjuic area, be sure to stop by the Joan Miro museum, it's well worth it for the art.

Also, I second the other people's reply about going up to visit one or all of the Salvador Dali sites in Figueres or nearby. Easily accessible as a day trip by train.