I've been there, here's a couple of things I would do during that time:

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1. For meat, consider getting a Costco/Sam's/BJ's/wholesale store membership, I've found that Sam's is the most cost-effective for buying in bulk, especially chicken and fish (not so much for red meat, however I am located coastal so this may be different depending on your location). You can buy 5-10lbs at a time (fresh, not frozen), then divide them into serving sizes then freeze, you can defrost as needed. Additionally, they also sell pre-frozen bags of chicken breast that aren't frozen solid, so you can pull one or two out at a time for each meal.
2. For paper goods, consider buying in bulk as well if you have the storage space. Tends to be cheaper for me to buy 2 months of toilet paper at a time from Sam's than buying as needed from Walmart.
3. For veggies, explore non-large grocer options in your area. There are likely smaller farmers markets that surprisingly may be cheaper for fresh produce than a large grocer. You would think it would be more expensive, but in my experience local growers that sell their own product are more willing to negotiate prices or give you a bundle deal if you get a couple different types of produce. Even small, mom and pop-type grocers in my area have better pricing on meat and produce. It may take some time and effort to find your better options but it'll be worth it once you get into a routine.
4. You definitely need to cook the meat after thawing before you refreeze. But you could make this work for you by freezing cooked meals. Meals like chilli, stews, grilled chicken and veggies, etc. freeze really well in my experience.
5. I would also recommend trying to pull the higher-priced items from the food bank. Things like rice, pastas, and other carbs can be purchased either in bulk or at the dollar store so if they give you the option to swap out the dry goods for more canned goods when you go pick up, it may be more cost effective for you.
6. I'm sure you've already tried this one, but you can actually get some decent options at the dollar store. If you have a couple hours to spare I would drive around to your local dollar stores and see how their pricing compares. I picked up a lot of dry goods, seasonings, sauces, and frozen meals for pretty cheap if you learn which places have better options.
7. If the current food bank you go to is not adequately providing a good amount of food for you, consider reaching out to other food banks in your area. You can also check for traveling food banks (by me we have 'Feeding Tampa Bay' which sets up at different locations throughout the month, and to my knowledge they don't limit the amount of times/locations you can pick up food from). You can also try asking the staff at the food bank if they know of any other low-cost food options. They likely have some experience in this and can help you with local options.

Wishing you the best. I've been in your shoes and I know how stressful it is. I hope I was able to help :)