"Off" can be an adverb, an adjective, or a preposition. In both your sentences, "off" acts more like an adjective than like an adverb.
As an adjective, "off" has several meanings. The Collins COBUILD English Dictionary* has several entries for "off" as an adverb and as a preposition, and two for "off" as an adjective. One meaning of "off" as an adjective is to indicate bad food, as in "The food is/ tastes off." Another meaning is to indicate something unacceptable, or wrong, or inaccurate, as in "His behavior is off." Your sentence "My guess was off" would fall into this category.
Sometimes even dictionaries don't agree on the classification of a word, though. The Collins COBUILD lists "off" meaning "have time off" as an adverb. and gives this example, among others: I'm off today. This is probably the meaning you are describing in your sentence "She is off today."
However, The American Heritage Dictionary of the English Language, Fourth Edition,** lists "off" as an adjective with this definition: "Absent or away from work or duty: She's off every Tuesday." Your sentence "She is off today" could also fall into this adjective category!
It would be interesting to discuss with your professor the classifications of "off" as they appear in different dictionaries. Even this statement from Quirk*** implies that there are differing opinions among grammarians about "off.":
"Off in The milk is off has moved into the adjective class for [some people]."
*Collins COBUILD English Dictionary. Harper Collins. 1995
**The American Heritage Dictionary of the English Language, Fourth edition. Houghton Mifflin. 2003
***A Comprehensive Grammar of the English Language. Quirk et al. Long