I'm not a SAHM, and my child was older when I began No-S... but I remember the days of being unable to eat, etc. Plus, I had real trouble with "No-S@home" for a while. So, I'm drawing on those experiences to offer suggestions. Use what you can, and toss the rest.
1) Always eat at the table, and make your kids do the same. This is good for you both. I have been SO grateful I instituted this rule early!! My child did have snacks when young, but they became part of the routine and gradually went away when he stopped needing them. He has very good eating habits now, in contrast to several of his friends barely eat at mealtime, but that's are always snacking on junk food between meals. I am so glad we didn't go that route. Added bonus: mess reduction - no food waste all over the family room.
2) If you can't eat with your kids, just have a drink and plan to eat at another time. You don't have to do it all together. Or, as Clarinetgirl suggested, plate your meal, but finish it afterwards (just try to keep it within a reasonable timeframe, so you're not really loading your plate to graze on all afternoon.)
3) Get out of the kitchen as much as possible. After meals, declare it "closed." If you need to get snacks together for your kids, prep them when you prep meals, so you can just get them at snacktime.
4) Empty your larder/pantry of things that you are most likely to snack on. Don't keep food on the counter (or at least, don't keep things you're likely to snack on).
5) To the best of your ability, plan meals you like (this is helpful for everyone, but I think it's especially important at stages where you tend to ignore your own needs, like when kids are little). And do plan ahead, so you don't fall into eating badly out of sheer exhaustion. Check the "intelligent dietary defaults" thread and come up with a few of your own, that you can throw together from items that are always on hand.
6) This is another general one that I especially recommend for you at this stage: Look for non-food things that make you happy, and incorporate them into your life. For me, eating was often a mis-guided attempt at self-nurture - food is comforting. But what I really needed might be sleep, or something to bring me joy. Buy flowers, indulge in scented bath items (neither of those take time), collaborate with your partner to get regular time for yourself, connect with friends... the larger your repertoire of "things that make me happy," the smaller the role that food will play.
7) Above all, be gentle with yourself. This is a demanding stage, but it will pass. When you fall off the wagon, dust off the crumbs and climb right back on. Give yourself credit for all you are accomplishing, and be your own biggest cheerleader.
Hope something here resonates. Best wishes!