Note that this answer only applies to combinations of singular words (i.e. not to "Meine Träume und Hoffnungen").
HalvarF pointed out that this question has been asked in German already.
The accepted answer to that question makes clear that both "drängt" and "drängen" would be acceptable. It also points out some reasons for using one over the other. In particular:
Bei formelhaften Wortpaaren ohne Artikel(!) steht das finite Verb vor allem dann im Singular, wenn das Wortpaar inhaltlich eine Einheit ausdrückt.
For formulaic word pairs without an article(!), the finite verb is in the singular mainly if the word pair expresses a unit in terms of content.
and
Je abstrakter die Bedeutung der Subjektteile ist, desto eher neigt man dazu, das finite Verb in den Singular zu setzen.
The more abstract the meaning of the subject parts, the more one tends to put the finite verb in the singular.
In this case, both apply.
Additionally, by using the singular, Goethe draws additional attention to the unity between "Dichten, Trachten, Hoffen und Verlangen", which might have been intentional. It implies (among other things) that him writing poems is the same as him desiring the addressee.
As Tilman Schmidt pointed out, there is also the fact that this is a poem. This offers additional freedom in terms of grammar but introduces limitations when it comes to rhymes and meter. "Drängt" fits into the meter and rhymes with "hängt" from the next line.