If you can get a good lawyer, I think you have a chance of getting this thrown out for an illegal search. I'm assuming of course, the shotgun was in fact loaded (though don't say anything about it here, save that for private conversations with a lawyer). I don't know if a public defender will do a good job arguing about it but ask him, hopefully the cops having no dash cam is worth something.
The
penal code also says this:
A person is guilty of carrying a loaded firearm when the person carries a loaded firearm on the person or in a vehicle while in any public place or on any public street in an incorporated city or in any public place or on any public street in a prohibited area of unincorporated territory.
I assume if you were on the highway, that's a "prohibited area of unincorporated territory" but you might ask about it.
If you do plea guilty, you don't have any aggravating factors (you don't have previous offenses, the gun is belongs to you, etc). As such you have a good chance of getting the minimum sentence, which is three months, and it's a misdemeanor only, so it won't be too much of a burden on your record. If you don't cause any trouble in jail, then you have a very high chance of being released early with probation. It's actually uncommon for people to serve the entire length of their sentence. So, try to stick it out, and you'll still be able to own firearms and get a job without having to declare yourself as a felon afterwards.
The court shall apply the three-month minimum sentence except in unusual cases where the interests of justice would best be served by granting probation or suspending the imposition or execution of sentence without the minimum imprisonment required in this section or by granting probation or suspending the imposition or execution of sentence with conditions other than those set forth in this section, in which case, the court shall specify on the record and shall enter on the minutes the circumstances indicating that the interests of justice would best be served by that disposition.
If I'm reading this correctly, it recommends the minimum sentence in your circumstances. You also might ask your lawyer if he knows what examples are of circumstances where the court deems to offer only probation instead of the minimum sentence. You may also be offered this in exchange for a guilty plea, which I would recommend taking if you don't have good chances on the illegal search angle.