Experts and BHG readers answer.
There is no easy way to get rid of nutgrass in lawns without herbicides. The Hi Yield chemical you used was probably MSMA, which works on a couple of kinds of nutsedge, but not others. There are two other chemicals that work on the other kinds, but they're harder to find and expensive. A good garden center or professional lawn care supplier would have them if you're interested. They're called Image and Manage.
However, if it's spreading as much as you say, then you probably have some drainage issues. You said you water in summer and have heavy clay soil, which suggests to me that you might be applying too much water. The clay, which doesn't drain too well, may be staying a bit on the soggy side, and nutsedge LOVES it soggy. So my suggestion is to cut back on water as much as you can. That will probably stop the spread. Wherever the nutsedge is growing in open ground (not near ornamentals or veggies), Roundup works fine. Other places, you either use the chemical option, or do your best pulling by hand. That's a lousy option, I know, but it's what's left if you rule out chemicals.