While Admiral Croft was taking this walk with Anne, and expressing his wish of getting Captain Wentworth to Bath, Captain Wentworth was already on his way thither. Before Mrs. Croft had written, he was arrived, and the very next time Anne walked out, she saw him.
Anne's out on the town with her sister, Mrs. Clay and Mr. Elliot - all of whom have ducked into Molland's emporium to get out of a bit of mizzling rain. Mr. Elliot has undertaken to see whether the Dowager Viscountess Dalrymple (whose carriage is down the road a ways) has room to take the ladies home. Having room for only four, and with two in the carriage, she offers to take two of the ladies with her. Elizabeth - er, "Miss Elliot" - is a given. And according to the usual pecking order, Anne ought to be next - she's socially far higher than Mrs. Clay - but of course Elizabeth has other ideas.
Who should Anne spot walking past the shop than Captain Wentworth? Anne is startled, and almost immediately contemplates running to the door - not to escape, but to follow our hero's progress down the street. She doesn't, of course; nevertheless, in a few minutes' time, the good Captain enters the shop with a party of his acquaintance, and on seeing Anne, he blushes. That's right: he actually turns pink in the face, and the only word Anne can find to adequately describe his countenance is "embarrassed." He is entirely discomposed, but he manages to pull his act together and come over to speak more sensibly with her after his brief initial greeting, although he never manages to be entirely comfortable .
Anne is pained (mortified is probably pretty close to the right term for it) when Elizabeth "would not know him." Austen is not saying that Elizabeth didn't know who he was. She recognized him, alright, and knew full well what the background and relationship was, but she did not curtsey or even nod her head at him - instead, she gave him the cold shoulder. And that is bad ton, ladies and gents. Badly done, indeed, Elizabeth!
When the carriage arrives, Captain Wentworth essentially offers his services to keep Anne dry with his umbrella and to hand her into the carriage, but Anne declines, as she is prepared to walk home. The good captain immediately offers her his umbrella, stating that he'd prefer to get her a chair - a conveyance carried by two porters on foot, but which kept ladies (and/or invalids) out of the weather. (I discussed them in more length - and with a picture - in this post from August at the Abbey.) She declines, as Mr. Elliot is coming back to walk her home, and Captain Wentworth is forced to listen to the ladies of the party as they all comment on what a fox Mr. Elliot is, and how lovely and superior (in a good way) they believe Anne to be.
Anne spends the next few days looking for Captain Wentworth, but cannot see him anywhere. "The theatre or the rooms, where he was most likely to be, were not fashionable enough for the Elliots, whose evening amusements were solely in the elegant stupidity of private parties, in which they were getting more and more engaged[.]" Ah, the elegant stupidity of private parties. What a perfect turn of phrase!
"The rooms", by the by, is a reference to the Upper and Lower Assembly rooms, which I described more fully in this August at the Abbey post. Anne has high hopes for the concert night (a concert held on a weeknight that was not taken up by the schedule assemblies), knowing that Captain Wentworth is fond of music. She is so keen to see Captain Wentworth, in light of both Elizabeth and Lady Russell having "cut" him, that she is willing to miss out on a planned visit to her friend, Mrs. Smith. Take note - she won't miss out on a visit with Mrs. Smith in order to make a certain visit to the Dowager Viscountess, but she'll do it for the possibility of seeing Captain Wentworth.
Mrs. Smith, you will note, says something rather arch about looking forward to tomorrow's visit, and it's clear from her comment about not expecting too many more visits that she thinks Anne's going to be taken away. But by whom? (Another fine example of a compelling page-turn on a chapter ending by Miss Austen.)