Here's what i did. I was a consultant before I started raising money, so I could carve out a chunk of my time for fundraising before any money came in. When the money came in, I jumped over full-time. If you're not independent yet, maybe you could shift towards contract work for the next 2+ months?
Also consider raising from friends and family first. This money can has the potential to come quickly if it's going to come at all. BUT don't do this unless your relationships won't be affected by the worst-case scenario (100% loss due to your obvious incompetence).
And I know your second paragraph is hypothetical, but if you have a product, active users, and co-founders, then you probably have time to talk to investors. Answer emails at night, and schedule meetings around lunch or personal days. Saying that fundraising is a full-time job is a bit exaggerated, at least for angel investment; I've been raising for 6 months now with some success, and it's maybe been a 20% of my time.