Some horary astrologers see the moon as the querent's co-significator. I think it makes the most sense to see it as showing the querent's emotional stake in the matter. Your 10th house moon could be interpreted as you having your eyes on the doctoral program as a career pathway. Indeed, a graduate program isn't the same as a liberal arts undergrad degree, but is a career pathway. The moon also works in a more active and abstract sense: if the moon is void-of-the-course (having made her last major aspect before leaving her sign) the querent's situation is unlikely to change. If it's in the via combusta, 15 Libra to 15 Scorpio, then the querent is probably also weakened.
A planet in detriment is in the sign opposite the one it rules. If your significator is in detriment, it shows a lack of strength. However, all is not lost. It may be in one of the angular houses or applying to a favourable aspect with one of the beneficial planets (Jupiter, in this case.)
I try to protect my privacy, but will say that I was in a field usually classified as a social science. I did work at research-intensive universities and supervised graduate students. My sense from "both sides now" is that a doctoral program demands that people develop a thick skin, and a willingness to focus almost entirely on one's research. If you have a plan for the type of research you want to do and which faculty members you want to work with to do it, that's important.
I wouldn't say that this is a bad time to apply. Usually these things are controlled by the department, as to whether students all need to start in September, or whether they can enter in January. This means that usually they have application deadlines. UW may handle things differently, however.
Finding the most auspicious date to apply for a program would fall under the heading of electional astrology. I find it to be tricky to do, as there are often some long-term hard transits that just aren't going away within the right time frame. But it doesn't hurt to play around with transits and see when you'd have the most going for you.
Incidentally, I began studying astrology ca. 1990 during a kind of dark night of the soul. I was in mid-career with a lot of favourable markers, but the center just wasn't holding. However, I kept astrology very much in the closet, as it would have not only jeopardized my standing within the department, but some of my colleagues would have feared that it would have jeopardized the reputation of the entire department. Hopefully your department is more relaxed about such things. I took an early retirement 11 years ago, and moved to a new location, so now I do what I want!