Nice report, as usual.
I was not aware that the National Cathedral had an entrance fee or was i just for a tour and/or elevator? This was my grandmother's church. I've been there several times (other than for a church service) and the only time I recall a fee was with my pre-COVID travelers' group - a member of the group got us a supposedly rare 'insiders' tour and I think it was only $25. And re: parking there is plenty of free on street parking within a couple of blocks at no cost as long as you follow the two hour limit.
My grandmother lived near Dumbarton Oaks and took us to the park many times. I thought I've done just about everything there is to do in WDC, but apparently I was not aware of the museum. Time for a visit, So, thanks for that!
I've never been in Hay-Adams. I do enjoy an occasional drink at Willard's Round Bar or at the St. Regis. Pricey, but a nice atmosphere.