up to our necks vs up to my neck

up to our necks and up to my neck both are adjectives.

Word	Noun	Adjective	Verb	Adverb
up to our necks	No	Yes	No	No
up to my neck	No	Yes	No	No

As adjectives, up to our necks and up to my neck are synonyms defined as:

Other synonyms of up to our necks include neck-deep, up to her neck, up to his neck, up to their necks, up to your neck.

up to our necks (adjective)	up to my neck (adjective)
deeply involved	deeply involved