This isn't what we do now, as our situation is different, but something I used to do:
Set my budget for the month for food. Took out cash to that amount, kept separate, and only ever used for food. No other money could be used for food and the food money could not be used for anything else! (just made things easier for me mentally) I did all of the food shopping so it was straight forward. I was never a shopping list/meal planner person, but I had a budget in my head for individual items - eg meat no more than £5/kg, get a mix (pork shoulder joints and & dark chicken meat largely as they were often cheapest, plus any offers or reduced goods). Buy large joints/packs if it works out cheaper and freeze. As I had a month's money in one go I was happy to buy more to save overall (ie no weekly budget). Buy frozen veg for basics we always like - spinach, broccoli, peas - to keep stocked up and as it was cheap. Buy almost everything on offer - eg stock up on gravies, sauces, cereals, ham... - cupboard items that have a long life, and everything else to freeze.
I found shopping on weekday mornings or late evenings easiest as the store was emptier; evenings were best as there were also more reductions. However this may not be easiest for your circumstances. If there is a day when one or two of your DC can be watched by someone else then go alone, as being alone is the key factor!
Currently I buy online, but I have a 'ceiling' price for meat and fish (as above, I go by kilo). I buy almost everything else on offer - if it's not on a deal, I tend not to get it unless it's key (milk and bread, mainly). I find this cheaper as we always make up meals as we go, but I wouldn't be happy saying, "This week we're having mince on Wed" when the mince is much more expensive than the other things I can get for that week, for example. Stock up the freezer, and any cupboard goods running low, again on offer. I get an order delivered once a fortnight and the bulk of our food comes from here. The first week it comes most of the perishables that I don't tend to freeze, like fruit, veg, milk, bread, comes with it. In the second week we will still have plenty of meat or fish (frozen) and cupboard stuff, so I have a look at what we need in terms of bread, milk, fruit, and veg and then get a small shop from a supermarket usually on my way home from work - I stick to the type of things we need, have a budget in my mind and my usual 'ceiling' prices, then I scan the reduced stuff for anything very good. I usually leave with the equivalent of one or two full carriers.
Well, that was long... 