My mother was in a movie theater when she went into labor with me.
How can I find out what movies were playing in June of 1946 in Seattle?
Do they have the Seattle papers on microfilm at the public library? Look in the classified ads.
Or the newspaper might have a searchable online archive for a price (less likely).
If you’re looking for influences on how you turned out, maybe you should check the movie listings for the first trimester. Could be that certain films noirs had an effect. 
No, I’m writing a little thing about my life. (Not for anyone else to read.) If I can find the movie I interrupted, I’ll start there.
The West Seattle Herald is on Google News Archive; on June 27, 1946 (not sure what date you wanted, so picked one at random), looks like these movies were playing:
Miss Susie Slagle’s
She Went to the Races
The Bell’s of St. Mary’s
Naughty Nanette
Getting Gertie’s Garter
Flame of the West
Three Strangers
Cinderella Jones
The Dark Corner
Crimson Canary
Classics, all of them. 
I’m not sure if it’s possible to browse for individual dates, but here are all the results for “West Seattle Herald” in June 1946:
Bing Crosby and Ingrid Bergman? That’s my guess.
Wow, I had never heard of the Google News Archive. Just when you think you have this Internet thing figured out it throws out something amazing at you.
Wow, that’s perfect! I found the movie! (Lover Come Back.) She’d told me the name of the movie years ago, but I couldn’t remember it.
The Shroud, you were very close. It was the 24th. However, she was in downtown, rather than West Seattle.