For the art museum: Definitely Miro rather than Picasso. The Picasso Museum is usually packed way beyond comfort. You need to push through crowds of people to get close enough to the wall to read the label beside the painting. It is no way to appreciate art. That was my experience in 2016, and it's obvious from posts on this forum that Barcelona as a whole has gotten significantly more popular since then.
The Miro Museum was a place I was able to walk up to and be looking at art just a few minutes later. There may have been 5 to 10 people in front of me at the ticket counter, but in general I think it's a place that doesn't require pre-planning. It's glorious in Barcelona to have a place for which you don't have to buy a timed ticket in advance. If you decide you definitely are going, you can check on whether they sell undated/non-timed tickets online to perhaps save a bit of time at entrance.
The trick to managing Barcelona as a tourist is to schedule one of your top sights first thing in the morning every day. Those are sights you don't have to worry about; you know you can be there on time without leaving some other sight earlier than you want to. If you aren't going to ascend a tower at La Sagrada Familia, there's a real benefit to going first thing in the morning. You walk into a nearly empty church, and it will be some time before folks from the next time slots are allowed in. If you go later in the day, you'll have holdovers with earlier tickets still clogging up the church. If you want to climb a tower, you'll only get a short-term benefit from choosing the first time slot, because your tower entry will be just 15 minutes later and you'll need to line up for that.
Parc Guell, being outdoors, doesn't feel quite as crowded as the other top sights. I didn't think the presence of a lot of other visitors significantly affected my enjoyment there. An early-morning visit would probably be a bit quieter, but check the weather forecast. I haven't been to Barcelona in April. Note that there's a long and mainly uphill walk to the park from the nearest Metro station, so bus, taxi or Uber would be good here.
Expect both Casa Mila and Casa Batllo to be packed to the gills. For me Casa Batllo was a bit worse. Visitors tend to follow the same basic route through those buildings (I used the audio guides), and I think they basically sell out all day long, so I'm not sure timing matters so much. Perhaps going about 2 to 2-1/2 hours before they close would find conditions just a bit better. But given the difficulty of guesstimating when you need to leave the previous location in order to hit your timed-entry time at Casa Mila or Casa Batllo, I think I'd schedule them first up on two of your Barcelona days.
One thing to keep in mind is that it appears the most crowded sites meter people in and out to prevent unsafe levels of crowding. Therefore, you might not get in the door exactly at the time shown on your ticket. Don't schedule things too tightly!
I second the recommendation for Sant Pau. So far, no one has reported needing to buy advance tickets there. The newly opened Casa Vincens is probably still flying under the radar. I haven't seen it myself so can't comment on how interesting it is compared to the other Gaudi buildings.
Most maps of the city flag a lot of modernista buildings, many of the them not open to the public but all with interesting exteriors. You can have an enjoyable time tracking some of them down.
MNAC is a large museum. You can usually walk right in. The modernista collection (furniture and decorative art) is very good, and there are magnificent medieval frescoes rescued from churches in the Pyrenees. It's free Saturdays after 3 PM, but if you're flying in on April 20, I don't recommend a museum that day.
It's mainly the modernista sights that are hyper-expensive. Barcelona has a lot of good museums. Choose what interests you.
