Traditional grammar would probably describe the paying &c clause as an “adverbial” modifying washing, arguing that it describes the manner in which the glasses are washed. If all you need is a grammatical pigeonhole to put the clause in, that’s fine; but it doesn’t really describe what is happening here.
Let’s simplify the sentence and compare it with other sentences which work the same way.
John washed his glasses, paying special attention to the frames.
John washed his glasses, hoping to see better.
John washed his glasses, standing in the kitchen.
John washed his glasses, listening to the baseball game.
There may be some justification for saying that the paying clause describes how John washed his glasses, and in that sense “modifies” washed; but what about those other sentences?
Traditional grammar will tell you that the hoping clause is an “adverbial of purpose”, implying that this modifies washed (since that is the only word in the main clause which may be modified by an adverbial). But semantically that’s nonsense: hoping doesn’t describe the washing, it is exterior to the washing, it causes the washing and causes John to persevere in the washing. It is John who hopes.
Traditional grammar will likewise tell you that the standing clause is an “adverbial of location”, again implying that this modifies washed. Again, this is semantic nonsense: the washing doesn’t stand anywhere, it is John who stands.
I can’t recall that traditional grammar has an explicit name for the listening clause; but it certainly doesn’t “modify” washed: it’s an action performed by John apart from and alongside the washing.
What all these clauses have in common is their attachment to John, not to washed. Does this mean that we should understand them as adjectivals “modifying” John?
I think not. The participles paying, hoping, standing and listening all stand in exactly the same relationship to John as the finite verb washed: John is their subject. We cannot call these “adjectivals” unless we are going to say that washed his glasses is an adjectival, too.
What's really going on here is that the paying / hoping / standing / listening clauses are new predicates, which share their subject with the main-clause predicates to which they are attached. We can paraphrase all of these sentences as two distinct independent clauses:
John washed his glasses. He paid special attention to the frames.
John washed his glasses. He hoped to see better.
John washed his glasses. He stood in the kitchen.
John washed his glasses. He listened to the baseball game.
Using the participle is a literary device for casting these new predicates as a subordinates of their main clauses. They’re not integrated into their sentences but tacked on to the main clauses as "supplements" (the term is drawn from the Cambridge Grammar of the English Language). The usual point of the device is to establish a non-coordinate connection between the two clauses: to “foreground” the action of the main clause against another “background” action. Sometimes the background action immediately precedes or follows the foreground action, but usually the two are simultaneous, as in your example. That may be paraphrased:
The AOA recommends that you wash your glasses every day. [As you wash your glasses you should] pay special attention to the frames and earpieces,where hair product and makeup tend to rub off.
