Meteor showers are named for their apparent radiant in the sky, so the Bootids would appear to radiate from the constellation Bootes, which in June is almost directly overhead from the Northern Hemisphere right after sunset.
If by "down and to the right" relative to Sagittarius, you mean they were coming from high in the sky and moving toward the SW shortly after midnight, then they would appear to be coming from nearer Lyra and Hercules. But, allowing for a slight difference in line of sight or interpretation, they could still have been coming from Bootes (which at that time would have been only a couple of hours west of the meridian).
From a dark-sky site on the Texas Plains early last week, we observed many more bright meteors per hour than is normal and many of them appeared to be coming from Lyra, though some were obviously Bootids.