Question
FOLLOW UP:  Actually, I think it is Bermuda Grass instead of Crab or Goose...it has white/red "vines" that are spread horizontally and interweaved with the turfgrass.

Answer
The trouble with overseeding in this case is that you don't want those Weeds to set seed.  Take a good look at them, then try the do-it-yourself identification of weeds at the Virginia Tech website:

www.ppws.vt.edu/weedindex.htm

and University of Illinois Interactive Weed Identification website:

web.aces.uiuc.edu/weedid

There is also the New Jersey Weed Gallery:

http://njaes.rutgers.edu/weeds/

The Illinois service is the highest rated page of its kind
on the internet but it requires a basic understanding of the
terminology -- or a lot of patience.  Both websites are excellent.  I have others, but these will do for now.

Some Weeds are annual, some are perennial.  If you are not positive about your i.d., head over to your local Cooperative Extension to get a confirm on your guess(es).  This is not something you want to mess around with.  As a famous Gardener once said, Know thy enemy.

If you see no sign of seedheads, and the Weed is identified as an annual, you can indeed sow directly over the dead Weed 'corpses'.  Just make sure your Seeds have good soil contact with the Soil.

Personally, I ENJOY removing every last trace of any Weed I see.  I compost everything, including the Seedheads.  They have to be well toasted before they go back to the garden, but I figure I am at least taking back a lot of what they stole out of my Soil.

If you give me your Zipcode, I will give you the contact information for your local Cooperative Extension.