The Acts of Paul seems to base the story of Paul’s conversion upon Galatians 1.11-17. A verbal comparison of the English helps us to see clearly possible influence.
The translation of the AP from Coptic is mine. Biblical passages are from the RSV.
Acta Pauli
ΠΡΑΞΕΙΣ ΠΑΥΛΟΥ/ Acts of Paul / Paulusakten / Actes de Paul
Acta Pauli 
The connections you note between Galatians 1 and AP IX hardly imply literary dependence, especially given the absence of any lengthy verbal correspondences. The common features between these passages could surely easily be explained based on oral traditions about Paul’s Damascus experience, combined with general Christian conversations about “revelation,” “the gospel,” etc. Similarly, it would be interesting to know the results of a TLG search on πολιτεύω.
Hi Julia: thanks for the comment. I hope all is well.
In the conclusion of my dissertation, I summarized which parts of the Corpus Paulinum were available to the author, based on strong verbal resemblances and historical information. It is very likely indeed that the author of the Acts of Paul had certain epistles available to him: 1 Cor, Gal, Phil, and 2 Tim.
Could features in the above text be explained by oral tradition? To be sure. Is that the best explanation? I don’t think so, because the case is strong that the author knew Galatians. Therefore, my view is that the historical information in the Acts of Paul is based both on oral tradition and on the Pauline epistles, but in absence of knowledge of the Acts of the Apostles. The likelihood that an author from the beginning of the 2nd cent. did not know Acts is also very strong, for apparently Acts had a less vigorous reception in the early decades of the second century than did the Pauline epistles and the gospels.
I am not questioning whether the author of AP knew Galatians, nor suggesting that he knew Acts. Indeed, I find the arguments in your dissertation convincing on these points. Literary dependence in this particular passage does not logically follow, however.
As an analogy, last week I participated in an interfaith discussion of the Lord’s Prayer. Because some Muslims present were not familiar with the prayer, we decided to base our conversation around Matthew’s text. Yet despite explicitly stating this from the outset, the Christians present consistently quoted the prayer as they knew it from liturgical contexts rather than Matthew’s form. Although they had read Matthew within the past half hour, their wording was not drawn from Matthew.
The very fact that multiple versions of the Damascus story exist suggests that this may have been a “well-known” story circulating orally as well as in written form. In short, AP IX certainly could depend literarily on Galatians 1, but it does not seem to me that you have sufficiently demonstrated that, given other possible explanations for the similarities.
Willy Rordorf wrote about Schneemelcher’s similar contention regarding author’s knowledge of Acts and his lack of dependence (Rordorf, “Was Wissen Wir”, 236):
I have to agree totally with Rordorf on this point.
Your example of the liturgical version of the Lord’s prayer is not easily analogous, since the liturgical prayer is not an oral tradition in any case: it is a rival written tradition which people have memorized.
I haven’t here presented my arguments–just a chart that shows verbal resemblances. I do believe that oral tradition has had an influence, but that if Galatians was available to the author, it explains a multitude of agreements–not just the ones in the chart above, but others of very substantive nature between Galatians and the AP, including the itinerary of Paul in the early episodes.
I am glad there is more proof to come, and I look forward to reading all about it in your commentary!