The subject is Trump. The structure is: verb + object + infinitive without to. Don't get confused by the fact that here the pronoun which substitutes the name "Trump" is an "object pronoun" (him). Actually, it's an OBJECT of the verb "watching" (which has "Washington" as its own subject), but at the same time it's also the SUBJECT of the CLAUSE GOVERNED BY THE VERB "WATCHING" (that is: "him savage critics). Then there is a coordinate subordinate clause that has him (Trump) as subject too (and inflame controversy...etc). Therefore, "him" is object of a clause and subject of another clause, but, according to the grammar rules of this structure, it has to be expressed in the "form" of an object. Note that the clauses with "him" as a subject end with the word "TWEETS", then there is the word "EAGER" which has an implied: "has been" before, so practically it is "has been eager", and this understood verb has "WASHINGTON" AS A SUBJECT AGAIN. So then it is like to say: "...and Washington has been eager for any sign he (Trump, here there is TRUMP AS A SUBJECT AGAIN) might be pivoting to a more presidential posture". I hope this analysis helps. Ask if anything is unclear.

It's a good question. This construction is like: "I saw the boys play/playing football in the street." At the time I saw the boys, the boys were playing football. "Washington" is the subject. Washington was watching Trump. When it was watching, Trump was savaging critics and inflaming controversy. I am not sure how to describe this construction grammatically. Probably, it is a kind of relative clause which does not need a relative pronoun.

Hi, I read again my answer and I think I must give you a specification about one thing: I wrote that "him" is object of the verb "watching". This is a logical explanation, but if I had to talk strictly in grammar terms, I should say that THE WHOLE SUBORDINATE CLAUSE , "...him savage critics..." (and "...(him) inflame controversy..." as well) IS THE "OBJECT" of the verb "watching". They are subordinate clauses that "act" as objects. But, as I told you before, "inside" the subordinate clauses "him" is the subject of the verbs "savage" and "inflame", even though it has the form of an object pronoun. I felt I had to give you this explanation so that you shouldn't have an incomplete information.

Washington is the subject of the verb "watching", and "him" (Trump) is the subject of the verbs "(to) savage" and "(to) inflame". Thus "Washington was watching Trump while he (Trump) was savaging critics and inflaming controversy... etc.

You wrote "critics and infame" in your answer, it's a typo, the word in the text is "inflame". Note that "inflame" is a verb, it seems to me that by the way you coupled it with "critics" you considered it a noun ("critics and inflame"), but maybe you didn't. Anyway the meaning is "Washington was watching him (to) savage critics and (to) inflame controversy with...etc.