Bobby, do you know what grass is in the lawn? Some varieties are a real PITA to kill, while others are easy and that would have quite an influence over my decisions. Is it seperate plants with fine roots, or are there roots running evereywhere? Yip, number two's PITA...
Without knowing your climate, I'd leave major planting till next season as I imagine things will be breaking dormancy about now.
Here's a rough sketch of what I'd do:
I wouldn't till at all,
especially if the grass is PITA. Aside from other stuff, tilling chops up those stolons and new grass sprouts from every piece.
If grass is PITA, I'd water the area deeply if it's not really moist and lay thick carboard or entire sections of newspaper, pre-soaked and
seriously overlapping.
Some people are very much against using newspaper/card like this; I'm not and it's your call!
If the grass isn't PITA, I'd skip the paper/card and go straight to mulching.
My mulch of choice would be many, many cubic feet of chipped trees. I'm hearing some stuff about chip spreading tree disease/pests in the US, so there's another caveat
Dump it on the lawn, say four inches thick at least.
Leave it till next year.
I haven't tried cover crops in new chip, and if it's PITA grass I'd very much avoid puncturing the card layer. If it's not, favas could maybe handle the conditions. Comfrey's a good one to get going, in
permanent spots. It's wonderful stuff, but once it's there, it's there...