Yes I do. Would the alternative be cue-ring (or cue-r-ing)? I definitely don't say that.
Hmmm - not exactly like cue-ring, because for me the 'u' sound is more like the 'oo' in 'book' than the 'oo' in 'cue'.
For me, during and jaw-ring (if such a thing exists!) would be pretty much homophones. Slight difference in the first consonant and the stress, but the vowel sounds would match.
Same 'oo' sound here for me as abovre, and the consonant would be a d-y (dyooring) rather than a j.