If I received unemployment benefits last year but did not receive a 1099-G and can't find it online with my state's search tool, should I still file it as income?

Yes, you should still file it as unemployment income and then exclude it. Please see this LINK.

I should add that the unemployment benefits were paid by my state government, not another source.

Yes, you would include the unemployment income, Have you tried to call the unemployment office? Here is a link to phone numbers of regional offices.

I've called my local offices but they either send you to a machine that keeps you stuck in a loop or send you directly to voicemail of which I have left a few already with no reply. 

The included link suggests filing as non-government unemployment assistance. The options I'm given are for union, private fund, or state employee benefits, all of which I don't qualify for. Should I just pick one and continue?

You will report it as unemployment.  None of the other options are correct.

• In the Did you receive a 1099-G? screen, select No, I didn't receive a 1099-G.
• On the next screen, Let us know if any of these uncommon situations apply, select I received non-government unemployment benefits that weren't reported on a 1099-G and Continue.

How to Report Unemployment if I did not get a 1099-G