I see your preference for beaches. We don't go to Europe to go to the beach, since we live on an island in Georgia with beaches very close by.

The Douro Valley is magnificent, we did a seven day river cruise there and loved it. Also, spent two full days in Porto. Make sure you visit Porto.

You can visit Evora on a day trip from Lisbon. Recommed 3-4 nights in Lisbon.

Nazare is very nice, we visited it on our way from Lisbon to Porto. Also, don't miss Obidos and Batiala (spelling?)Cathedral.

Haven't been to the Algarve.

I've not been to Sines so can't comment on it. I think with 6-7 days to see sights, I would stick with 2 spots. My personal choice would be Evora and Lisbon. I think Evora would be a nice spot to recover from any jet lag. There is a great Wines of Portugal shop where you can try all the wines of the Alentejo region for about a euro each, and you could also drive to some wineries around the area. Evora itself is a lovely walkable town.
Villa Vicosa is nearby as well to day trip to. If you do go there be sure to find out when the English tour is. They only let you in during tours ( We had the Portuguese tour, which was apparently very amusing if you spoke Portuguese).
So I would do 2-3 nights Evora and 3-4 in Lisbon. You can day trip out to Cascais for beaches and Sintra for more sites.

Then you could decide if you want an extra night for Rome and Amalfi.

Way too many 2 night stays, which is really only one day in that destination. I agree that two destinations in a week is plenty. Especially since there re great day trips from both Lisbon and Porto. There are beaches within a short train ride of Lisbon. Lots of nightlife, culture, food wine, etc.

And, if you stick to Lisbon and Porto, you can avoid renting a car - and don’t want one either of these places. A car would be good in the Douro valley but thats really the only place it might be needed.

Save the Algarve and some of the smaller places for a future, longer trip to Portugal.

If beaches are your priority, then I would suggest that you stay on the coast for a week in somewhere such as Foz do Arelho or Sao Martinho do Porto, as I think Nazare is overrated. I have stayed here and in a week, I have had a day in Lisbon and have visited Obidos, Tomar, Fatima and Alcobaca plus the coastline.