Geez, doesn't it seem like I could put the mayo packs in my clear plastic bag with my shampoo?

Absolutely! There's a company that makes peanut and almond butter packets that are under 3.4 oz, and we've brought those and little jelly things in a baggie before. Not for the plane, but for a snack once we were there.
As long as you have passed through security, they don't care if you have liquids on the plane.
I know you're talking about domestic flights, and this happened on an international flight going home, but it just goes to show you can never be 100% sure... My husband was coming home from Japan, he got through security and was near the gate. Bought a bottle of water for the flight, a big one, because it was going to be clear through to SEA. As they were preparing to board, there was a last minute, at-gate, security check. And they refused to let ANY liquids onboard. He had to toss his brand new, unopened, purchased minutes before, water.
So they *can* care.
You should be good with a sandwich, but you cannot carry a jar of peanut butter or jelly through security.
And if you put PB in your checked luggage, you're almost certain to have TSA check through it. DH's last two trips were to places where he could get a kitchenette in his hotel room, and with easy shopping. So he bought PB each time. And brought the half jar home, because hey, half a jar of PB is hard to throw away! And both times he's had notes from TSA in his checked bag, LOL. This last time he had a bag of ground coffee right next to it, which is also a known hiding spot for things that might make drug-dogs excited. We were surprised he didn't have two notes from TSA!
I'm sorry for being nosy, but I've just gotta ask because the curiosity is killing me. Why would you be traveling with cans of vegetables in your carry ons?
People do odd things...I was right behind a family that was bringing things like baked beans in their carryon. Strangely, they were let through. I asked the TSA guy about it, asked if the cans were let on (since they do have liquid in them, and the liquid isn't specifically measured, and that's the whole problem with snowglobes, right? (though I didn't mention all of that in my question to the TSA guy)), and he said that he just asked them to take them out and put them into a bin in the future, so TSA didn't have to search the bag after the xrays showed up.
If you can ignore the comments by people and just read the blog, the TSA blog is interesting, to see all the junk people try to get through security! Also it lets us know the ridiculous things people SAY to airline employees and TSOs. People are weird!