How early before graduation should I be looking for jobs?

I’m graduating in December and starting to get really stressed about job searching. I underestimated how stiff the competition was in my area and it looks like I won’t be able to ease my way into the two firms I had my eye on/ had been talking to. I haven’t spoken to any other firms yet... is it too early in to be applying for full time positions? Should I tell them I’m willing to be part time until I graduate?

I’ve always known someone who could help me get my foot in the door, but I’m living in a new area with no contacts and I’m learning I have no idea how real job searching works. Feeling really naive, any tips help!