When my daughter had it, we at first tried the RID, which didn't get all of them. Like you, we didn't want to use it again, so I looked on the internet for a natural cure and found one that worked great.
First you slather the hair in mayo. It has to be real mayo, not Miracle Whip. Apply it like you would conditioner, but a little heavier. This smothers the adults and helps break down the bond that the nits have on the hair. Then wrap the hair with Syran Wrap, around the forehead, etc to really seal in the hair. You can also use a shower cap, but it needs to be fairly snug around the head. Take a warm towel and wrap that around the head. Keep it that way for around 2 hours. I just read other sites that say overnight, but the two hours worked for us. I only heated the towel twice I think. My daughter watched a movie during this time, and it didn't bother her at all. After this you need to use an apple cider vinegar rinse. This step will sting a bit where you have been bitten, but it is a natural antiseptic. The vinegar helps to rinse out the mayo, and also helps get rid of the bond with the nits. Wash with your shampoo. Then find any remaining nits. They should come right off. Her hair was super soft after this, and lice free.
As far as everything else, wash all bed stuff in hot water and dry on high heat. Wash any combs or brushes too. Combs can be put in the washer, though I wouldn't dry them. If you can, wash the back of the couches, etc. where everyone sits. Put stuffed animals that they sleep with in airtight ziplock bags for a day or two.
You can also use an inexpensive olive oil instead of mayo. That will also help smother the adults.
Best of luck!
H.