Director @ Startup: How To Give Notice with uncertain job offer?

I am seeking advice on how to best proceed with this unique situation.

I am working for a startup in an executive-level director capacity that has funding until the spring of next year. Personal issues (read: pandemic uncertainty) have prompted me to seek employment with a more established firm. I successfully negotiated an equivalent position with an established national company and even managed to get a raise!

The new company wants me to start part-time next week and then go full time the following week. I got the official offer letter last Wednesday. The offer is contingent on a successful HireRight background check, but they say not to sweat it if HireRight has any issues verifying employment. I have no criminal record and have most of my financial documents to prove old places of employment.

The problem: I do not want to give notice until the background check clears. I am running out of time to give a two-week notice. Since this is a startup, can I offer to contract with them and work a few hours here and there, given my executive capacity? I care about the company and the people in it, but I cannot deal with the uncertain future.

Any tips on how to manage this? I feel as if I'm in an odd situation.

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