There does not seem to be a way to search for an album in the Add to Album dialog. When I am adding photos to an album, I am dragging the album I want to use to the top of the list of albums, right before I start adding more photos to the album. Then "Add to album" will show this album as the first album in the list.
To move the album to the top of the list, tap "Albums", then "My Albums", "Edit". Now you can drag the album you want to the top of the list and start adding mor photos to it.
Working with albums is not well supported for large photo libraries on the iPhone. My Photos Library is still moderately sized, with 55000 photos and videos, just 240GB. Half of the storage on my iPhone 11 Pro Max is still empty, but I have more than 1000 albums. Having to scroll this long list is a pain. My work-around is to structure the list of albums with folders. This is possible, if we are syncing the iPhone with iCloud Photos with a mac. Photos for Mac allows us to group the albums with folders and subfolders, and the iPhone will show the nested folders when browsing the list of albums in an "Add to Album" dialog. Without a Mac to organise the albums in folders it is very hard.
