I've done some of the typical stuff like writing down key points and rehearing answers ahead of the interview. I also seem to do ok with the coaches I've occasionally enlisted.
Love to hear from someone who has solved a similar problem.
I am also curious how and how much do most people practice (esp the ones who aren't exactly glib talkers)
I know this doesn't help that much right now while you're looking for a job, but it really helps to practice all the time.
For a more practical solution: I go for a 30 minute walk every morning, and during that walk, I often talk aloud about whatever is on my mind. Very often, I'm practicing a pitch that I want to make to someone else, or a speech I want to give, or a witty, snarky comeback that I should have given when Bob insulted me in front of the cashier at Starbucks. I do it every day, and I've noticed that in my "real" life, I stutter less and I come across as more persuasive than I did years ago before I started doing this.
So to answer your last question: every day for at least 10-30 minutes :)
Write down at least 5 value-add stories and practice reciting them smoothly in advance.
Preparation is key!
If the interview process is dehumanizing or badly conducted, this is a signal about the company/department itself sharing the same characteristics. If I don't see bad intent, I also assume that the interviewer is trying to do their best for me to succeed. Their aim should be to get the most out of every candidate and not to pressure someone into failing.
This may sound irrelevant to the fact that you are finding it hard to describe your achievements, but I think that the mood of the interview matters a lot. A positive mindset on your part can relieve a lot of the tension and with less tension, there's more bandwidth to think.
Practice
And practice some more
Have a handful of "good stories" ready to pull out
An interview is a great time to deliver your elevator pitch (https://writing.stackexchange.com/a/33505/1317)