Of course, check the flue per all of the good advice you've received above. You need a solid flue for safety. And, woodburners tend to be an occasional use item in that part of CA, so you see a lot of "alternative" and downright unsafe installations. But if you still have a problem, I've been there.
I lived down the road from you in Livermore for a while. I had a conventional fireplace with the same problem. I had a few theories about what caused it. You can see if you think any apply to you.
1. The tendency for the weather to be 60 degrees +- 10. If everything is the same temperature, there is little positive draft, and maybe a negative when the fire is out and the masonry is cold, a day or two later. If the stove isn't sealed (and modern stove primary and secondary intakes generally don't seal) you're going to have a smoke smell when draft goes negative for whatever reason. The reasons my draft got negative follow.
2. A typically leaky California house. If there's no really cold weather, finding and fixing leaks isn't a priority. Though there were no obvious problems in our house, a good wind outside would cause helium balloons to blow around inside the house.
3. A tree wall perpendicular to the prevailing wind that created lower pressure at the back of the leaky house, along with the chimney flue being toward the front. Combine that with a leaky house and it's breeze in through the chimney, and out through the back wall leaks.
4. If it was cold (okay maybe 35-40!), and the chimney hadn't been used, stack effect would have cold air enter through the cold chimney and exit heated through the leaks in the taller second story, no breeze required.
Yes, my fireplace had a damper, that I closed, but it didn't seal perfectly, similar to the modern woodstoves with the always open primary and secondary air supplies. My stoves here in MT will give off a flue smell sometimes during the brief times during the summer when they're not in use, but it's not much of an issue since it's summer, the windows are open, and ventilation is good. If not in use during the winter, they have a good positive draft because of the warm inside air and the cold outside air. They're wasting good warm air up the chimney, but they don't smell inside.
I solved my problem in Livermore by simply not using the fireplace, except on the very rare occasions that I decided that a fire was worth a week or two of smoke smells. Annual heating bills were less than $40 anyway, and since it was a fireplace, it made the house colder overall when it was in use. I realize that isn't the solution you're looking for, but I decided not to install an insert in my fireplace (though I would have liked the ambiance) because I concluded that unless I could seal the drafts, it wouldn't have solved the problem. Though I probably could have come up with a burner that would fit, and that I could seal up somehow when not in use, a little ambience, and offsetting less than $40 a year in heating bills, never got the issue far enough up on the priority list to get done.
If you have a sealed flue, the problem will stop if you can seal the stove when it's not in use. The thing is, that's impossible to do with the design of a lot of inserts. If you can access yours, masking tape, plugs, plastic wrap, or whatever works applied to ALL of the draft inlets should solve the problem. Other than that, my experience says the smell pretty much goes away a few weeks after the last fire, or when you light another one.
In that part of the country, I think a woodstove is fun (or not), and good to have as a SHTF backup, but a heat pump makes sense.