Lots of good suggestions. But, as is common with these types of threads, also just a LOT of suggestions period (many not falling into the categories the OP mentioned)
So I will limit my suggestions to what I would consider the “kick off” for the category. If you like these, then you can move on to some of the others.
Action
War movies:
“The Guns of Navarone” and “The Dirty Dozen”. Great escapist, impossible task that could turn the tide of the war (WWII). Both start out a bit slow, but are worth the big finales
Dramas about important issues:
“12 Angry Men” is probably the best for this. Great study of how each person’s experience shapes their view/opinion. And is like “homework” should you ever participate in a jury.
Also mentioned “Mr. Smith Goes to Washington” is a nice fable about a “common man” taking on politics, and holding onto his principals.
Horror. Not suspense horror, but supernatural horror:
“The Exorcist”. You have probably already seen a number of movies that out-scare, out-special effect this movie. But this scared the crap out of me (and many others).
Disaster! :
“The Poseidon Adventure” kind of kicked off the star-packed, “epic” disaster film genre. Compared to today’s “earth ending” disaster movies, these movies seem so “contained”. But viewed as the survivor’s tales, these earlier ones (also “The Towering Inferno”) are more about the human experience than the big special effects (now all CGI’ed).
Political thrillers:
“Fail Safe” comes to mind (especially since you liked “Dr. Strangelove”). This is a more realistic take on the possibility of nuclear annihilation/cold war mentality. I don’t know that many today realize just how tense things were during the cold war.
Scifi with good science:
“The Andromeda Strain”. Awesome science, and a refreshing take on a first encounter with extra-terrestrial life, which holds up today.
“Marooned”. Before the real-life events of Apollo 13, this movie did a great job of depicting the inherent dangers of manned space flight.
I’m ashamed to admit that I’ve never seen a Sean Connery Bond movie:
Bond movies are great escapism. I’d recommend “Goldfinger” and “You Only Live Twice”. If you like those, then go for “Dr. No” and “From Russia With Love”