I doubt Anne has any physical health problem. Jane Austen gives little direct description of Anne's features. I think Sir Walter sees only himself and Elizabeth in bloom while everyone else is plain or old !
My impression is Anne outwardly suffered some loss of her looks due to the traumatic experience of loing the man she loves. Of course, at age 27, she is older than other JA heroines.
Outwardly, Anne has lost her bloom but retains some inner strength and self-reliance.
Anne at Uppercross has the mental resources to listen and accomocate the Musgroves. True, it was a trial to her nerves listening to them discuss Capt. Wentworth (ch.6) but I don't recall her having headache or really depressed. Mary actually feels her ailments lessen with Anne as companion- I doubt Anne
could be capable of this if she felt so poorly as Mary. ;)
I think Lady Rusell was being practical regarding city heat and potential effect on Anne.
JA wrte how she disliked the heat and white glare of Bath.
Also, in summer I believe Georgian cities were dangerous with dirty waters, open sewerage & outbreaks of disease. People who coud afford too, often went to the country.