Your strawberry and veggie patches are almost done for the year where you live, so it seems to me it would be best to tackle them next spring. For now, I would cut down the weeds as close to the ground as possible, but not worry about getting rid of them altogether.
In the spring, when the strawberries (and weeds) start coming back, you'll be able to hand pull the new weeds much easier than you can now, because they'll be small and easier to pull. And right from the start you should mulch your strawberry bed heavily. Straw is the traditional mulch for a strawberry bed, and you should put it down in a 12 inch layer between the strawberry plants. This will make finding a safe weedkiller a moot point.
In the spring, I'd also till or dig the entire veggie garden under. You mentioned that you have raised beds - if there are sides, you may be stuck doing this by hand. Again, you can mulch between veggie plants to prevent the weeds from growing back when they come up. Hand pull what does come up. The early you get them, the better. If you do need to use a weed killer next year, you could try concentrated vinegar, which is safe and effective. Keep in mind that it will kill any plant you get it on, so you'll need to be sure you get it on the weeds but not your veggies.
It isn't terribly easy to maintain a chemical-free lawn. Acres and acres of the same plant growing in a monoculture is just not natural, so you're working against millennia of evolution. Your safest option is to hand pull the weeds. If there are more weeds than grass, you could smother the lawn or till the lawn under and start from scratch. Next safest would be to spot treat the weeds with the concentrated vinegar I mentioned. Other chemical- free options would be boiling water on the weeds, using a propane weed torch, and Burnout 2 (a citric acid/clove oil product). Any of these will kill the grass along with the weeds, so you'll need to do some work (spot reseeding) to help the grass grow back after the weeds are gone. If you decide to go with a conventional herbicide, your safest bet would be to select one of the broad-leaf herbicides on the market and spot treat them.
To be able to limit the amount of herbicides you have to use going forward, make sure you're using good lawn care practices. Mow your grass long so it is stronger than the weeds. Three inches is a great length. Work to improve the soil in your lawn. Water it deeply about once a week if you aren't getting rain.