If my child graduated from college in May 2021, do they count as a full time student from Jan-May?

Yes.  The definition of a student for tax purposes is as follows.  

IRS definition for Student:

A full-time student is a student who is enrolled for the number of hours or courses that the school considers to be full-time attendance.

To qualify as a student, the person must be, during some part of each of any five calendar months of the year:

1. A full-time student at a school that has a regular teaching staff, course of study, and a regularly enrolled student body at the school, or
2. A student taking a full-time, on-farm training course given by a school described in (1), or by a state, county, or local government agency.

The five calendar months do not have to be consecutive.

Q.   If my child graduated from college in May 2021, do they count as a full time student from Jan-May?

A. Yes. If the school classified him/her as full time.  Any part of the month counts as being a student, even if graduation day was May 1st. On the other hand, if your student finished up his degree (Jan - May 2021) requirements by taking fewer courses, than a full time load, he would not be considered  a full time student from Jan-May and would not meet the requirement for a Qualifying child type dependent.

Graduation year 

If he/she was a student (under 24) for at least 5 months and lived with you for more than half the year, and did not provide more than 1/2 his own support for the whole year, you can still claim him as a dependent. Be sure he knows you're claiming him, so he doesn't claim himself. He can only be claimed once. But, he can "file taxes" without claiming himself.

The real question is who should be claiming him in this "transition" year to adulthood. You two have to agree on who is going to claim his exemption. Each should do their taxes both ways and see which way the family comes out best.  Even then, you have to meet the rules. The rule is that a child of a taxpayer can still be a “Qualifying Child” dependent, regardless of  his income, if:

1. he is a full time student under 24 for at least 5 calendar months of the year (graduating in May usually means you meet the 5 month rule)
2. he did not provide more than 1/2 his own support  (scholarships are considered 3rd party support and not support provided by the student). 
3. lived with the parent (including time away at school) for more than half the year

So, it usually hinges on  "Did he provide more than 1/2 his own support in 2021.

The support value of the home you provided is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants. IRS Publication 501 on page 20 has a worksheet that can be used to help with the support calculation. See: http://www.irs.gov/pub/irs-pdf/p501.pdf  (page 15)