It's difficult to provide a very brief phrase here that will accurately capture the action being performed. I can't think of a verb offhand that covers this motion. "Dangling" suggests being fully suspended. "Depending" might work, but is very vague without further explanation. If this image is important to your scene, consider inserting it prior to the conversation. Then, "dangled", "depended", "hung", and so on would be adjusted in the reader's mind to match what you just told them. Or, "...as he continued to fall, then catch himself...".
Your question has a great deal to do with your own personal narrative style. "Balancing" doesn't really fit the situation, but many verbs or short phrases would work provided you set the image in the reader's mind first.
"Leaning" does sound like a workable option. If you feel you've overused this word, don't forget that a previous usage might be the better one to change so you can use it here without exhausting the word .
Don't consider previous text to be "written in stone"; I revise entire previous scenes for all sorts of reasons. Occasionally, for exactly this reason; I too hate overusing a word.
Consider rewriting this section in an entirely different way; that often provides answers for these awkward examples where the writing just doesn't "work".
Let us know what you come up with.
Student or Learner