You probably have nothing to worry about. Any dust that was released into the house air is gone by now. Even wood dust, can be carcinogenic, but to lose sleep over your casual exposure (if there was any of consequence) is obsessive.

Have the home air tested, if you think that will set your mind at ease. Give up smoking, if you engage in that dirty habit. And, consider avoiding trips in the family automobile on interstate highways...now 'that' is a hazardous and life-threatening activity.

Your one-time exposure is probably of no consequence to the health of you or your family members.

Well, I was able to find a small piece under a cabinet and send it out for testing. Turns out the backing is 15% chrysotile asbestos. Trying not to obsess--- though I am steam-cleaning my rugs, wiping down hard surfaces and throwing away the vacuum I used to clean up the dust the saw made... Anything else I should do?

I know Armstrong used asbestos backing on the designer solarian in my old house. I don't like to hear of it being cut with a circular saw. I think you could ask your city building inspectors for the hazardous material department. I went to them when I sold my old house and asked about specific asbestos hazards and disposal methods. I'm pretty sure they would help you but be careful you don't trigger expensive remediations.

Well, what's done is done so we can't do anything about disposal or remediation. We may have an air test if we can afford it. Thanks for the advice--I almost wish I didn't find out about this.

Yeah, too bad you didn't know beforehand, but it is highly unlikely that you or your family members will have any adverse physical consequences due to casual exposure.

You should use a good dustmask when doing any kind of work like sanding or sawing. When you're done, vacuum yourself and the site well. Wash your clothes.
The most danger your kids will be in are the effects of lack of exercise and obesity.