How can I kill weeds without killing the whole lawn?

Look for a weed killer that doesn’t target your lawn.

when I had St Augustine grass, it was tough to find one, as it’s similar to some weeds.

You can apply a liquid based Lawn weedkiller using a sprayer or watering can. Usually only one application is needed to kill weeds for good. You can kill most weeds including dandelions, daisies, clover, plantain, common mouse-ear, white clover, black medick, lesser trefoil.

Lifestyle Homes is right that you need to purchase a weed killer specifically for lawns. This will be chemical based. The best 'natural' way to get rid of weeds is to physically keep pulling them out. Yes, it is part of the ongoing high maintenance that a lawn requires.

The weeds are telling you that something is out of whack with your soil. You may need to aerate or your ph is off. Mulch your leaves into the soil, cut your lawn higher and don't bag it, mulch it.

Preen for lawns

https://www.preen.com/products/preen-lawn-weed-control

If you're referring to crabgrass, it's best dealt with before ground temps are above 50°F. Once over 50°F, there's a product I believe is called Tenacity, used by lawn professionals, but available to homeowners via the internet. You Tube has detailed videos, using this product.

If you have purslane, it's indicative of soil that's too acidic, and you may need to lime your yard, typically in the Fall, in the Northeast.

Wild daisies generally indicate soil that is overly compacted/heavy clay.

You may need some soil analysis tests, performed by a County Extension Service, usually affiliated with larger colleges, or similar.