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Title. Is it possible to work corporate counsel without going the big firm route? Are there any public interest gigs or gov't work for regular law grads that line up with corporate work?

It is, but opportunities to do this are fairly rare, as companies want you to come in with some substantive training/don't want to have to train you to do your job.

That said, I know of more than a handful of tech companies in Silicon Valley (i.e. not Apple, HP, etc., which have formal entry-level hiring programs) that will hire people right out of LS. But you generally need a connection to and/or professional experience with the GC to get these.

Tl;dr - This is possible, but opportunities are rare.

If your goal is to have a long career working in-house, I would not skip the big law step, even if you have an opportunity to do so. Put in at least a couple of years in big law. You'll unnecessarily limit your in-house advancement otherwise.

The real question is how soon can I leave biglaw

Yes, it is absolutely possible, just not that common. Some in-house legal departments will even hire directly out of law schools (though this is more common with big companies that have huge legal departments, training programs, etc.). Someone I know graduated from a T25 school and went directly in-house to IBM with no biglaw or firm experience at all.

Some of the research triangle companies do straight to in-house. SAS/ Toshiba