Deductions & credits

Here's the criteria to claim a 'qualifying child':

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a US citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
  • There's no age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

So if your child is over 19, you're correct you can't claim him as a dependent.

For a 'qualifying relative':

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a US citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $5,050 in 2024.
• You provided more than half of their financial support.

You would qualify for an exception to the 'lived with you entire year', but If you don't pay over half of your son's living expenses (paid by the state), you probably wouldn't qualify to claim him.  However, since you are POA and may be paying the facility from funds that you are responsible for, you could consult a disability expert to see if that qualifies you as 'providing more than half their financial support'.  In that case, you could  claim him as a dependent for a $500 deduction, and claim his medical expenses if you itemize deductions.  

However, whether 'your' lodging would still be included is still a question.

Here's a Dependent Interview from the IRS that may help, and some further discussion on the issue of Claiming a Disabled Child as a Dependent. 

@sstace