legacyfamilytreewebinars.com ran two fairly recent webinars that might be extremely useful to you, "Weaving DNA Test Results into a Proof Argument" by Karen Stanbury, and "Finding Missing Persons With DNA Testing", by Diahan Southard. You would need to buy at least a short-term subscription to the webinar site to view them, but in your situation, I suspect you'd find it worthwhile (I have no affiliation with Legacy Family Tree or the webinar site).
I think you've started out on the right path by (a) testing at AncestryDNA, the largest DNA subscriber base around, and (b) submitting your sample data to GEDmatch.com. @Jaclyn is correct that the next step is to find close maternal relatives to get tested, so that you can use the details of those matches to filter out more distant maternal-side matches. In practice, it will take more than one, possibly many. Obviously, the best option would be to get your mother tested, but from what you say that seems unlikely. The next best choice would be to essentially try to reconstruct her genome (or that of her close ancestors) using first your own siblings (if any), then your mother's siblings (if any), first cousins, etc.
Using the chromosome browser at GEDmatch, you can compare your sample to your close maternal relatives and identify specific DNA segments that map to your maternal ancestors.
At that point, what you're looking for is "shared matches" on AncestryDNA who match you, but not any of your maternal relatives. When you find some, you may need to persuade them to also upload their sample data to GEDmatch. If the segment(s) you match overlap with segments that you share with maternal relatives, but they don't match those relatives, then most likely they're matching on the other half of the half-IBD (i.e., your paternal DNA).
If, on the other hand, they match segments that you haven't established maternal-side matches for, you can't be certain, unless you have enough samples among your maternal relatives that the lack of a "shared match" with any of them indicates a high statistical likelihood of a paternal-side match.
With enough such paternal-side matches, you may be able to form a hypothesis as to a most recent common paternal ancestor, where your paternal matches fit in that tree and, using the ISOGG DNA relationship tables, where your father might fit. You can then run your theory past your mother and see how nervous she gets.
The DNADetectives Facebook group is also useful, but be aware that, like most such groups, there's a tendency to over-emphasize the success stories, which are actually the far end of the bell curve. The likelihood of finding a close paternal cousin match on any of the commercial DNA sites is statistically fairly small (yes, a few people do, but not many). What you're much more likely to see is matches against somewhat more distant cousins, but more of them (possible many more).