Have any of you settled with an insurance company without a lawyer?

You can. They are likely to call you. They will offer you a check and ask that you sign a waiver for any future claims. Just make sure that you are confident (medical exam; reasonable length of time) that no future injuries will crop up.

I have done it and had it work out OK.

Yeah, I did it for a minor accident when an elderly man rear-ended my car because he wasn’t paying attention. Unfortunately I had just come from having minor surgery on the some skin on my back and the area started bleeding again. :-P Everything ended up being ok, but his insurance company freaked out when they heard that and offered to cover all the doctor’s expenses plus extra. So you shouldn’t have a problem, especially if the other person admitted it was their fault.