For me, once't in an oakie vernacular would be a contraction of once and it:

I'd grab a beer once't finished.

but of course that's just me and I left the Central Valley a long time ago and am not a professional linguist, I'm just telling you about my perception of the vernacular from where I grew up.

It’s an archaic version of whence.

I was teaching my native English speaking students German and explained that woher and wohin are like whence and whither, and the looks of utter unrecognition made me realize I was doing this. I tried for a moment with “thence and thither? Hence and hither?” And 4/7 knew “hence,” but that was it, and it’s not really used in the same sense as the others, anyway (I mean, it is, but it’s metaphoric and most people saying it probably aren’t thinking of it in that sense).

If I were to affect an oakie, I would produce “onest” to mean something like “singular” or “unique”, or possibly “solitary” or “lonely”. I would mean it as a superlative of the ordinal "one" and it would be pronounced slightly differently.