First off, hats off for carrying even though your family doesn't approve, because I think that they would be pretty quick to change their minds if ... and I don't want to paint in somber tones here, but hypothetically let's say the psycho that was accosting you had carried out whatever sick, demented fantasy he may have had planned had he not seen your Sig.
I honestly can't stand people who think violence, rape, abductions, etc. are things happening in far away places and could never happen to them, its just absolute ignorance. My mother in law detests guns thinks only cops and robbers carry them. Trust me, its a pretty sensitive issue for me as she spends a lot of time with my 18 month old, taking her to the mall, lakes and other places where they could be targeted. I bought her a can of mace and its still in the package it came in, which really bothers me because I take life seriously and am aware of the different risks that are taken every time you walk outside.
I have always been comfortable carrying guns as I have been shooting since I was 8 yrs old. I have been jumped and robbed, beaten with a baseball bat, brawled in my dooryard with guys that came to my home and believe me, that s*** aint gonna fly anymore, especially since I have a family of my own now and am responsible for taking care of them.
You have a responsibilty to defend yourself and your family and it sounds like you are taking it seriously. Don't let others discourage you, they are obviously not taking responsiblity for protecting you and your family so it is your job to get comfortable, and do what you know is right. A lot of what it comes down to is exposure, those who do not know a lot about guns or how to use them, go based off of a lot of what they hear and see on TV, I think you may be a little apprehensive due to the fact that it is somewhat new to you and that's normal.
I think the more you shoot your gun and clean it and get to the point where you can assemble/reassemble with your eyes closed, the more comfortable you will get. The more and more you carry it I think that you will find that you will feel naked without it, that's how I am. I am never more than 3ft away from a loaded pistol.
My wife is kind of new to the idea but she is really receptive and willing to learn everything I have to teach her and I couldn't be happier. Basically, just shoot it, train with it, and don't go anywhere without it and after a short time you will have no quams about carrying it. Good luck and don't let other peoples misconceptions discourage you from doing what you need to do for yourself and for your family.
Remember, the only reason they do not carry is because ultimately they are not responsible enough to, but you have taken that on and that's admirable. Don't let the bad stigma surrounding guns get to you, gun grabbing conservative liberals intentionally paint that picture of people who carry guns but they are not responsible for you and yours.
When push comes to shove and you're put in a situation you're gonna want options, so don't hesitate.