We've started recently with this too and are on a pretty tight budget. Our family is also breaking a lot of costly and not so healthy habits right now with these changes.
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I've focused on getting the best quality meat/ we can find and afford from local sources and then adding what we have to from the grocery store after that.  With meat, I try to buy a lot of the less expensive cuts.  I found for example that oxtail from my favorite farm is pretty affordable and that I get a lot of bang for my buck with it since I cooked the bones a second time for more stock.  Also, I found that cutting ground meat with organ meats like ground liver (in the food processor) or ground beef  heart both ups the nutrition of our meat and stretches it further. Meatloaf with ground liver in it is so yummy and my husband never did figure out what was different about it.  Also, I can make at least three meals out of a whole chicken (using the meat in stews or casseroles) usually and have the carcass for stock so a whole chicken from a reputable farm, while expensive goes a long way for us.  I also like to get sausages from our favorite farmer.  They are a little pricey, but I can brown it and freeze it in small amounts to use as pizza toppings or tossed with some homemade soaked pasta or in some stews, so it goes a long way.  We also find it useful to keep some canned Alaskan wild salmon in the house to use with pasta or on sandwiches.
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 Often, I find I can freeze half of what I make as our main dish and thaw it out for dinner another day which keeps us from having so many wasted leftovers and means that when I've had a hard day I'm less likely to send DH to the store to pick up something quick.
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We supplement our meat use with lots of legumes, either soaked or sprouted and then cooked and added to a dish made with bone broth. Once a week  I like to sprout rice and lentils and then cook them with bone broth, coconut oil and as many veggies as I can, seasoned with some thyme and sea salt.  I eat this for lunch most days.  It fills me up, is relatively inexpensive and has lots of good nutrition in it.  It's fantastic with some sauerkraut or cortido on top.
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We get unhomoginized but gently pasturized local milk and I make yogurt or kefir from it. I also use it for whey and cream cheese (which my son loves for his breakfast/lunches). Â I'm scoping out raw sources, but I'm not sure they are in our budget atm. Â For cheese, we are buying small pieces of raw cheese from the grocery store, and I buy one dozen eggs at the famer's market and then try to find the best we can in the grocery store for any additional eggs we need during the week since two dozen eggs at the farmer's market is not affordable for us. Â We eat the farmer's market eggs first and save the remainder of the grocery store eggs as spares for the following week.
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Fermenting veggies is wonderful and lots of fun.  I've been able to get 5+ pound heads of cabbage at the farmers market recently for a dollar a head and have been making lots of sauerkraut and cortido.  We haven't tried kimchi yet but it's on our list.  I also buy lots of carrots but mostly organic from the grocery store and they go into lots of things.  For  veggies I set a mall budget for the farmer's market each week and get what I can there.  Often I buy what is the best deal that week.. and then figure out if it's something I can put into a stew  or rice and bean dish or ferment.  I've actually been surprised with how much produce we can afford from the farmer's market this way.  What ever else we need produce wise we get at the grocery store but honestly it's not much these days.  In terms of the fermented veggies I aim for eating some with every meal.
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We also figured into our budget, that since I'm making kombucha now, Â and eating traditional foods, I have kicked my soda and fast food habits and that saves us money each week too which we can put towards buying healthier higher quality foods.
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DH has been pretty resistant to much of this.  The only fermented veggie he will eat is ginger carrots and only if I put them in salmon salad. He also complains about so many beans and lentils, but he has seen me drop a little  weight and gain a whole lot of energy in the past month and he's starting to see some value to it and refrains from bringing junk into the house now.  He has also really has a hard time with increasing our fat intake, but again, he sees it working so he's not complained too much.