stick welding old thin header tubes?

Wow!
I am a journeyman welder and if someone said that a 5/32'' 6011 is the smallest rod you can get, they are out to lunch.
You can get them in 1/16'', but i would not use a 6011 for this at all, a 1/16'' 6013 would be my choice if stick welding was the only option.
Using a copper or brass backup strip would be a good idea.
Personally, i would tig weld it or gas weld it after the joint was cleaned well and all rust removed.
Forget flux core or innershield. Mig would work if it was very clean.
Welding any rusty exhaust pipes is just a crapshoot anyway......it will never look pretty, just seal and hold.
My advice is to take it to a muffler shop and get them to scab it together for you. they might even do it for a 6 pack of beer or pop!
Want it to look better after it's welded?
That's what they make grinders for.