In Which I Face My Failures

So the other day was an Errand Day, not to be confused with a Shopping Trip Day, which at least has the potential of fun. Shopping Trip Days usually hold the promise of a trip to at least one large store with makeup departments and home decor departments and kids' clothing departments in which I may wander at my leisure provided neither child pitches a fit or fills their pants. Whereas Errand Day, by definition, involves lots of small trips to boring destinations, with multiple gettings in and out of cars and carseats, and is not something I look forward to.
The first stop was the DMV. Now, I don't know about your town, but the DMV nearest to our neighborhood is grim at best. The floors are dirty, black flecked linoleum; the walls are a dirty white, as are the chairs lined up before the dirty white counter. There are many dusty fake plants sitting around, and the walls are "decorated," inexplicably, by large, postcard-like photos of random waterfalls and beaches at sunset which I can only surmise to be ancient castoffs from a travel agency. The women behind the counter literally never smile, have uniformly terrible haircuts, and don't own a flattering shade of lipstick among them. They speak in voices devoid of emotion. It is a place I loathe. I feel gritty when I leave, as though a layer of the place's dust and defeatism has settled on me while I was waiting in line. I was doubly dreading the trip this particular day because I had to take both kids with me. I was anticipating having to keep Addy quiet and well behaved in the essentially library-like atmosphere while hoping Eli didn't awake and start rooting around discontentedly, forcing me to fumble with blankets and shirts and flaps of my bra while pretending to be totally cool and un-self conscious about breastfeeding in public.
But the whole thing went off without a hitch- I got the kids in, settled Addy in a chair with a box of chicken nuggets, dutifully approached the counter to take my number, and was told I could go next! The paperwork went quickly, and a nice old lady talked to Addy the whole time and kept her from running around. Eli slept, and the lady at the counter smiled when I left and called me honey! A miracle which filled me with hope.
The next trip was to evil Wal-Mart, as there continues to be no Target in a fifty mile radius. I only had a few things to pick up, and I was anticipating no problems as Wal-Mart trips generally go smoothly. But it was here that my bubble was burst. Within five minutes, Addy was so desperate to be out of the cart she was about to throw herself overboard, so I finally released her, and then chased her around the aisles for twenty minutes. And no, I couldn't put her in the seat-belted front part, because I had to clip Eli's carseat there, and no, I couldn't have put Eli's carseat in the back because our Wal-Mart's carts are tiny and there would have been no room for both a carseat AND my purchases.
I finally managed to get to the check out, after helping her put about fifty Halloween greeting cards back in their appropriate slots, and we headed towards the parking lot. I staved off a near tantrum at the soda fridges beside the counters and was almost home free. Then we passed the stupid rides and arcade games and all that crap. Side rant: Getting out of your average discount store is like a freaking gauntlet of temptation for kids- candy bars, impulse-purchase novelties like nail files and Pez dispensers and Kleenex purse packs. Then the inevitable bucking broncos and swaying cars and glittering glass boxes filled with mounds of tantalizing stuffed animals just waiting to be plucked up with a mechanical claw. It's like the store is booby trapped!
Now, I patiently let Addy play around in the rides for a little while, but it was only delaying the inevitable. Suffice it to say, Addy was not pleased when I finally decided it was time to exit the yellow plastic car and go home, and all hell quickly broke out. Oh, how I hate these scenes- the yelling, the stares, the attempting to physically manipulate the child back into the cart without prompting any calls to Children's Services by concerned onlookers. I was sweating even in the autumn chill, Eli was grunting hungrily from his perch in the cart, and a largely pregnant woman was watching me with wide, fearful eyes. Finally, carrying a screaming and flailing Addy under one arm and pushing the cart with the other, I marched resolutely towards the parking lot. Only to be tapped on the shoulder by a kindly older woman, who was holding one of Addy's light up Dora the Explorer tennis shoes, apparently kicked off in the scuffle. I shamefacedly accepted it, and Addy calmed down, her face still flushed and tear streaked, distracted by the appearance of her beloved shoe. If this hadn't happened, I'm pretty sure I would have had to continue on to the car carrying a hysterical child over whom I clearly have far too little control.
This worries me. I need to get a grip on my kid here before she gets too big to carry, basically. I realize that this scenario was merely embarrassing, but it brought up a real concern for me. I need her to listen to me, at least about important stuff like not running away in parking lots, etc. And I am not a spanker- not staunchly and adamantly and self-righteously, but it's just not me, and since I can't ever see myself doing it with any confidence, I'm pretty sure it would simply be ineffectual at best and confusing at worst, so let's start by saying that that particular disciplinary tactic is not part of my repertoire. However, putting a kid in time out is challenging when said kid is going limp and sliding off the chair while screaming.
Can anyone tell me what I am supposed to do about defiance and tantruming? It's embarrassing, but it's also concerning on a more fundamental level. I am all for having a loving and playful relationship with my kids, but I'm getting a little worried that there's a bit too much play and not quite enough respect in Addy's and my relationship. Have I already failed- is it too late to reign things in? Help!

Edited to add: Is everyone else taking their kids to fun fall festivals but us? Everywhere I go in the blogosphere I encounter adorable pictures of children with giant pumpkins! Stop showing me up! My idea of a fun trip with the kids is talking about the names and colors of the produce at the grocery store.
Oh and also: My cheesecakes turned out be-yutifally and I was outrageously pleased with myself. I have great plans for pumpkin cheesecakes this Thanksgiving. I was so proud of myself I went on to make fudge, which I have never attempted before. It was pretty good, too- not fabulous, but good.