Cheryl Lynn

I have found New Jersey Tea to be variable in when it germinates.  I  do a boiling water bath, then cold stratification or winter sowing.  Out of 20 seeds, I may get two sprouts after a couple of weeks of warm weather.  But when I've dumped out the soil from the "dud" pots and used them for other things, I will find NJ tea seedlings in those pots after a month or two.  
I have also planted out those not very strong looking seedlings, and then seen them come up the next year looking pretty good.
I hope your NJ tea gives you some nice surprises!  But I would suggest marking where you planted 'cause it may not look like much early on

What have you observed about frost in your yard so far?  Meaning this spring, have you been getting frost in the part of the yard where you dug?  
I'm in WI, also on sand, we have been having a slow spring which has given way too many opportunities to study frost/non-frost microclimates.   Where I am, even a little interruptions (e.g. a tall shrub or or above ground object that warms up or last years peas on a trellis will affect where it actually freezes.  But my yard is open and affected by cold air coming in on a wind.  I thought I remembered that areas with trees stayed a little warmer than completely open areas.    Long story short, you might get frost, but probably not everywhere.  Maybe plant some frost sensitive plants like beans or basil in a few weeks that will show you where it gets cold first.  

The warm season grasses may need the soil temp to be 70 rather than low 60's.  I've been successful with side-oats grama and bottlebrush grass, growing them outside in WI (zone 4b) after it warmed up.  The rye is a cool season grass and shouldn't be bothered.   Prairie Nursery lists a cold dry stratification for prairie dropseed rather than the no treatment suggestion from Prairie Moon.  I had scant germination on that, but my seed was wild-collected, so I don't know the quality.  I have grown Culver's root (again, outside) and it did nothing for a long time, and then all those tiny seeds geminated at once and I had a lot of babies to separate.  

Is your grass growing flat to the ground or more upright?  If flat, it could be crab grass.  Crab grass is an annual, but the plant makes gazillions of seeds that are happy to germinate once the soil warms up.    If it is more upright, then possibly nimblewill which is a perennial grass.  Both plants are warm season grasses (meaning they grow like gangbusters after it gets warmer.
The smothering techniques with cardboard/mulch should work (probably better on the crabgrass since the nimblewill might creep out from underneath before it croaks).    A long term strategy is to make sure your natural area has more cool season sedges.  These will shade & cool the soil and take up space before the the opportunistic warm season grasses take over.