Buy a bag of ice and cooler.
Buy a cooler but get your ice free at the hotel. Bring a small cooking device of your choice with you to fix a meal before you leave in the morning or after you get to your next destination. Pack canned fish.
You can get a hot plate for $20.00.
Also grocery stores with salad bars.
I did a road trip on the cheap a couple of years ago and the cooler was one of the smartest things I did. I stocked up on all kinds of yummies from Market of Choice and ate like a queen on beautiful cheeses and prosciuto, etc., while I was on the road. Motel 6s (which had a really nasty reputation when I was young) are now family friendly and every one I stayed in had a fridge, microwave, and all but one had free internet (that one charged a whopping $2.75); a few had pools. I also brought my own coffeemaker (didn't need it too often) and my own coffee because even with the general public moving toward better coffee, I hate getting stuck having to drink "lobby" coffee or the nasty generic coffee they have next to the coffeemaker in the room.
I wasn't primal at the time, so I ordered pizzas when I was in need of a hot meal (odd to think of it now). But for restaurants, I think I'd stay away from chains as much as possible. Ask the concierge or front desk person - they usually know good stuff. Then do the best you can. Depending on the trendiness of a given area, you might even be able to find places that serve grass fed meats. If not, stick to lean cuts of animal and some local fish. Depending on your budget, you may or may not be able to be perfectly primal on the road. Don't sweat it, do the best you can, and enjoy the scenery!
Amazon has hot plates. As for using a cooler and ice, there are coolers you can plug into your cigarette lighter adapter if you don't to use ziplock bags and deal with melting ice.
go camping instead and cook over fire.
Have lunch in a public park with grills and cook enough for dinner and breakfast.
Since I'm the kind of person who can live for a long time on "cold cuts," I'd recommend them - especially if you can stock up on better quality ones. HB eggs. Veggies. Just as you can find small restaurants in almost any town, you can almost always find grocery stores. Be creative - if you see a sign for something like, "Boiled crawfish - $3/lb," on some stretch of road, change your plans for that night and pick some up.
You also asked about storage. Baggies/ziploc bags will leak if the ice melts too much. Plastic containers with screw-on lids work better. But your best bet is to make sure to empty the cooler and get new ice in it every night when you pull off the road to sleep. The only problem I had with food getting wet was the night I was so exhausted that I pretty much just fell into bed. Stuff was floating a bit the next morning.
Not really for a main dish, but you can cook sweet potatoes in a microwave (assuming the hotel provided one). They're easy to transport. Also not a main dish, but if you did have a microwave you could bring those "steamer" bags of frozen vegetables. Or cart along a small blender for smoothies.