Acting amorously refers to behaving in a loving, passionate and romantic way. There are various ways to express this behavior in language. You could say that somebody is being affectionate, flirting, courting, wooing, romancing, caressing, embracing, cuddling, snuggling, petting or smooching, for instance. Alternatively, you could describe somebody as being smitten or besotted, infatuated, lovestruck, enamored, captivated or attracted to somebody. All of these words and phrases describe somebody who is acting in a romantic manner. By using different synonyms for the word "act amorously", you can add variety to your writing and express this behavior in a different way.