Massive Johnson grass invasion

I have a vegetable garden area of 15 raised beds which I have allowed to get totally overwhelmed with Johnson grass.  It got hot; I got busy.  It's just so shameful, but after I've wallowed awhile in self-recrimination, I need to figure out what to do now so that I can reclaim the space for a vegetable garden next year.

So here's my plan - What do you think?

(1) Spray Round-Up on the Johnson grass--pretty much the whole area.  I know, I know--I definitely lean strongly organic, but I'm reading that Roundup is the only way to win even the first skirmish with the J-grass.  It's too late to solarize, and (high school teacher) I don't have all kinds of time to dig and  pull at this point, even if I had the most meager hope of making a dent in the stuff.  (One might say that a high school teacher should never even try to have a garden, but...I like it.) 

Question:  If I spray with Round-Up, should I try to take the grass down with a weedwhacker first or just spray the tall...uh, really tall...stuff?  Yes, it has gone to seed.  So embarrassing.  You should see it.

Question: If you think I should take the weedwhacker to it, should I just try to hack it off halfway?  All the way to the ground?  I don't know that much (obviously) about how Round-Up works.

(2)  Take apart the raised beds and till up the whole area.  Then try to rake out some of the rhizomes and consign them to a pit of fire along with the (hopefully dead) stalks.

(3)  Start laying in as much organic material as I can find (chicken house offerings, fallen leaves, etc.), let it sit there through the winter, (till again?) and try again.

Question:  Till again in the spring?  Or just start planting and hope for the best?

Gardening experts, where are the flaws in my plan?  Should I just try to sell the house and move?

Thanks in advance for considering my poor plight!  I know I count on you! - Rita G./Cass County