engage vs sign

engage is not a noun while sign is a noun.

Word	Noun	Adjective	Verb	Adverb
engage	No	No	Yes	No
sign	Yes	Yes	Yes	No

As verbs, sign is a hyponym of engage; that is, sign is a word with a more specific, narrower meaning than engage:

Other hyponyms of engage include featherbed, fill, engage, ship, contract, sign on, sign up, rat, farm out, job, subcontract.

engage (noun)	sign (noun)
	structure displaying a board on which advertisements can be posted
	a public display of a message
	a gesture that is part of a sign language
	any nonverbal action or gesture that encodes a message
	a character indicating a relation between quantities
	a perceptible indication of something not immediately apparent (as a visible clue that something has happened)
	a fundamental linguistic unit linking a signifier to that which is signified
	an event that is experienced as indicating important things to come
	(astrology) one of 12 equal areas into which the zodiac is divided
	having an indicated pole (as the distinction between positive and negative electric charges)
	(medicine) any objective evidence of the presence of a disorder or disease

engage (adjective)	sign (adjective)
	used of the language of the deaf

engage (verb)	sign (verb)
get caught	be engaged by a written agreement
consume all of one's attention or time	communicate silently and non-verbally by signals or signs
give to in marriage	mark with one's signature; write one's name (on)
keep engaged	make the sign of the cross over someone in order to call on God for protection; consecrate
engage for service under a term of contract	communicate in sign language
hire for work or assistance	place signs, as along a road
ask to represent; of legal counsel	approve and express assent, responsibility, or obligation
engage or hire for work	engage by written agreement
carry out or participate in an activity; be involved in
carry on (wars, battles, or campaigns)