A sentence or two. Enough for them to know whether it is their area — not your whole history.
Optional, and it helps a great deal when you can be specific. Skip it if you are not sure.
Specific times beat "flexible" — flexible means they have to ask.
The sentence that does the most work
“If you aren’t taking new clients, is there someone you would suggest?”
A therapist with no availability has nothing to offer your enquiry and frequently doesn’t reply at all — there’s no good version of that message. Asking for a referral gives them something useful to send in one line, and a colleague’s recommendation is worth considerably more than a directory listing.
A surprising proportion of people in this city get seen this way rather than through the person they originally contacted.
Send eight to twelve
This is the correction most people need. The instinct is three careful messages and a wait; the response rate doesn’t support it. Full practices triage enquiries and many clinicians can’t reply to everyone. Why nobody calls back.
Keep a list of who you contacted and when, because after a fortnight you won’t remember. One follow-up after a week is reasonable and frequently works.
Two things to leave out
Your whole history. A first enquiry is a scheduling question, not an intake. Two sentences is right; two pages gets skipped.
Apologising for contacting them. Extremely common in these messages and it makes them harder to answer. You’re asking a professional whether they have availability, which is an ordinary question.
Before you send
Settle the money question first, so “how you plan to pay” is accurate — the four questions for your insurer. And widen the geography, because any California-licensed clinician can see you by video, which changes the availability picture more than anything else you can do.