What is the technical musical term for singing and sobbing at the same time? Can I say singhiozzio?
The usual way would be with a gerund, like crescendo or diminuendo.
So my bet is on singhiozzando, which seems to be confirmed in Rigoletto, p. 14:
This word is defined in the OnMusic Dictionary:
A directive to perform the indicated passage of a composition with a sobbing effect, especially in vocal and string music where it can be marked by a strong portamento.