Those months you mention are good. My advice is to do the walk with someone, not alone, for safety.
As for which month to walk, they all have their positives and negatives, depending on what your priorities are:
May: Generally cooler, more rain, not as busy with school groups and university students. Everything will be very green.
June: A little warmer than May, but still likely to have some rain. It will still be very green. Schools are out, so there will be more school groups and university students.
October: Similar to May with regard to the number of people, but probably less rain than May or June though, but similar temperatures.
Here's a good site with year-over-year temperatures and rain fall for Lugo, Galicia, which is close enough to that part of the Camino to give you a good idea of what to expect weather-wise:
The last 100 km is the busiest section, so even if you walk solo, you won't be alone. Plenty of people around. And the crime rate along the Camino is pretty low. I wouldn't hesitate to walk alone myself.
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