Yay, Determined!! Sounds awesome - you can do it!

You're completely right that it involves planning ahead and some extra effort, but it is totally worth it. I don't have a large budget for groceries either, so I can definitely relate to that, however I am a vegetarian so I am not purchasing the most expensive food (meat), and most of my main foods like beans, lentils and veg are pretty cheap. One thing I do though is religiously comb through the weekly grocery flyer and then sort of plan what I'll eat for the week based on what items are on sale in the flyer. When it comes to fruits and veg, I pretty much only buy what's seasonal or items that are in the flyer, and each week there is pretty much always 5-10 veg and 5-10 fruits on sale in the flyer. So if one week the veggies on sale are cauliflower, carrots and beets, those will be my veggies for the week instead of buying broccoli or peppers. If apples are on sale, I'll buy those instead of oranges. Usually there is one type of nut on sale, so I'll have almonds one week, pistachioes the next, etc. I always stock up on shelf-stable goods like rice, pasta, canned beans, oatmeal etc whenever it's on sale for a good price. I only have a couple of staples that I buy every week regardless of whether they're on sale like eggs, yogurt and my favourite brand of wholegrain flax bread. If I find that I have money left over I will use it to "splurge" on something a bit special like a high-quality cheddar or parmesan or some special out of season fruit like mangoes, so that I don't always have to feel like a total penny-pincher who can never buy anything special.
You might also be surprised to find how cheap it is to make your own soup rather than buying it canned. What I do for soup is keep tupperware containers or large freezer bags in my freezer, and then fill them up with any food odds and ends like broccoli stalks, carrot and potato peelings, the bottom of the celery and the leaves of the celery, scraps of onion, mushrooms that dried out, etc. If you eat meat, you'll also want to add any chicken bones or meat bones. Once I have several baggies full, I empty them into my crockpot (or just a normal pot on the stovetop is fine), fill it up with water, add a bit of olive oil, salt and pepper, a dash of tamari, parsely and a bay leaf. You can add any other seasonings or herbs you may have on hand. Let it simmer for a few hours, strain and you have your stock, essentially the base for a delicious soup. Add any veggies you wish, some protein (beans, chicken, ham, whatever you want), a can of crushed tomatoes if you want it tomatoey or curry powder or cajun spices if you want it hot, etc. Healthy, homemade and yummy! It sounds time-consuming, but it's really not because you just save all the scraps as you go for the stock and you can just throw anything in to make the soup and use canned or frozen veggies, canned beans or meat if you wish to save time on chopping. It's also extremely cheap - the stock is virtually free because you're using scraps that have already been paid for and then just the cost of a few veggies and protein. I divide it up into individual containers right away and you can freeze them or just keep in the fridge to eat throughout the week.
Regarding the sandwhich, I don't think there is anything wrong with eating a sandwich as long as it's got some protein and is on whole-grain bread (which you've got covered). If you want to mix things up though, you could do "sandwiches" on whole-wheat pitas or do whole grain tortillas with some salsa, veggies and meat. Bagged coleslaw tends to be really cheap and is a good way to add some veggies and crunch onto a sandwich, plus it tastes really good with mustard mixed into it. As an occasional treat I like to do a variation on the standard peanut butter sandwich - I take a small whole wheat tortilla, spread it with a tablespoon of natural peanut butter, put a small banana in the middle, drizzle it with a small amount (like a teaspoon) of honey and roll it up. Zap it in the microwave for a few seconds so it's just a bit gooey. If I'm feeling really crazy, I add some raisins too. It's definitely not low carb, but it's reasonably healthy, very filling, unboring, and is a nice sweet fix. I also like egg salad lettuce wraps. It's just two hard-boiled eggs with honey mustard and a tablespoon of natural mayo. You can usually buy lettuce wraps or romaine hearts that are good for rolling up sandwich fillings like egg salad in. I know lettuce wraps instead of bread sound very "diety" but they are surprisingly amazing with the egg salad because you have sort of the decadence and creaminess of the eggs contrasted with the crispness and freshness of the lettuce. Two eggs with a tablespoon of mayo, a bit of honey mustard and the lettuce wraps is under 300 calories. Other good easy lunch options are cottage cheese with fruit stirred in, hummus (very cheap to make your own) with assorted raw veg and whole-grain crackers, a chili or stew (more filling than soup), and if you're not concerned about the carbs in potatoes, a baked potato (sweet or jacket or red) with a small amount of cheese, sour cream (just don't overdo the sour cram and cheese) and lots of veg like broccoli or baked potato topped with black beans and salsa is amazing too (I don't actually bake them, just do the potatoes in the microwave).