This might be a very beginner question but what's the best way to clean a sport route with a very steep overhang? On lesser overhangs when being lowered off the anchors I would always clip a draw to my belay loop and then clip the other end around the rope to guide my down so I could pull my quickdraws off. However on a steep overhang like this one it doesn't seem to work. Gravity takes effect and I still end up getting lowered more or less straight down.
All you have to do is have an attentive belayer stop lowering once you become level with the draw and you pull yourself in. Find a nearby handhold to take tension off of the draw and unclip. Keygate biners make things easier. Do keep yourself clipped into the lowering rope with another draw as you described.
Mike Minson has the best solution. Following is a pain unless you will not fall off. When you fall following it is not always possible to get back on. Better to tram in and clean on the lower.
indeed, especially in the southeast where the routes tend to be excessively steep and you usually are gassed by the end of the day....or sometimes "way before" then..
To add to MM's post, if it is getting supersteep, flipping upside down and pulling yourself back in hand over hand is sometimes "helpful". It also can take a "team effort" between climber and belayer to "bump" the cleaner back in at critical moments and keep the slack from decking the climber at the first bolt....lots of variables to pay attention to...