I would say yes, 'handicapped' is older usage, more outmoded, and more likely to be experienced as thoughtlessly offensive, than 'disabled.' I would support marking it [dated] in LEO, and I support the suggested additional entries.
Yes, 'persons with disabilities' ('... cancer / Down's syndrome / schizophrenia / AIDS / PTSD' / you name it) is the most recent and most PC way to put any of these categories, putting shared personhood (positive, inclusive) before categories associated with disease (negative, divisive). People writing longer texts, especially in academia, will be expected to know that.
(You wrote to Andrew Solomon? Wow, that's like the time Marianne wrote to Michael Mosley. Yay you. (-: )
However, 'persons with ...' is usually too long for signs or concise informational contexts. That's why the wheelchair symbol is often used in place of any words at all, and when there needs to be a single word, that word is now indeed 'accessible.'
So yes, the least bad word for people is still probably 'disabled' -- in AE as well.