Hi Kelly,
I walked from SJPdP to Fisterra this year, starting on April 1st and arriving on May 5th, also as a solo woman .
Regarding clothing, it wasn't nearly as cold as I expected it to be. In the end, the perfect combination was a quick-dry running t-shirt, a lightweight fleece jacket and a very thin running jacket. Even on cold mornings, I was always warm enough by adding a woolen hat and a thin but broad scarf (which could double as a wrap/poncho in the evenings when I needed to wash my jacket).
I definitely recommend take a sleeping bag! Most albergues did provide blankets, but in some cold nights it was nice to have both.
Trekking poles - I didn't make up my mind whether or not to take them until I arrived in SJPdP. I ended up buying a pair there and boy, I was glad I did! Especially on slippery paths downhill, they provide much needed stabilty. What worked really well for me was a pair of fingerless cycling gloves - I got a blister on my palm from the wrist loop and the gloves helped (and were warm enough for me).
As others already said, there's no need to worry about finding an open albergue. Even on days when I didn't arrive until 7:00PM, there was always a bed to be found no problem.
Weather-wise, the first half was mostly cool and perfect for walking (the Napoleon route was closed when I started, but the Valcarlos route really is most lovely - if you take this route, just remember to save some energy for the last part where you go steadily uphill for quite a while), I got a few very wet days in the middle section (no more than 4 or 5 in total though) and towards Galicia, the most beautiful spring/summer weather emerged with some 28°C days so I feel that I got a bit of everything and loved it. Well, maybe I didn't love the day when I was ankle deep in mud and almost swam into Léon quite as much at the time. ;-)
All in all, I think you have chosen the perfect time of year - I loved my spring Camino and would go again at that time of year again without thinking twice!
Buen Camino and enjoy!