This is definitively how time works in Watchmen. It's also how it works in other stuff like Interstellar, Slaughterhouse-Five, a ton of other fiction, and, I'm pretty sure as far as science can tells us, the real world.
Here's a person, on a Monday at, let's say, 12:01 am. In bed, asleep.
Here's that person through the rest of the week at 12:01 am. Tuesday, up working late. Wednesday, on the couch watching a movie. Thursday, going through the fridge for a snack. Friday, out at the bar.
Here's that person at noon throughout the week. Monday, in the office working. Tuesday, across the street eating lunch. Wednesday, at the doctor's office. Thursday, in the bathroom, Friday, sleeping in.
You plot these points and every single point in between, all of which are the facts of a persons existence at that given moment, and you have a person's path through space-time.
Taken from the moment they're born until the moment they die, you have the entire person's existence mapped in a blob whose shape/form is defined by the things that person did(does/will do). Call it their Space-Time Existence Blob.
As far as we know, these blobs are, have always been, and will always be fully formed. There isn't an objective timestamp for what moment the universe is at. The past, present, and future exist simultaneously at different points in space-time, the same way that Monday, Tuesday, Wednesday, Thursday, and Friday co-exist on the graph.
We can't see our blobs. We can only see the infinitesimally small slice of moment we call "the present". Dr. Manhattan, on the other hand,
can see his, and the shape of his Space-Time Existence Blob is even informed by the fact that he can see it, but he's every bit as equally powerless to change it as us. Because there isn't a difference between the future, past, or present. They're all simultaneous points in his STEB. As someone who is constantly aware of every single one these points, Dr. Manhattan is a hodgepodge of what we consider the past, present, and future, but that doesn't give him any special power over the shape of space-time. The structure of time is fully formed.
Changing the future is literally the exact same thing as changing the past. It's changing the shape of your unchangeable blob. In the case of the tachyon cannon, Manhattan only knows about it because it happened, and if he stopped it from happening, it wouldn't have happened, so he wouldn't have known about it to stop it. That'd be an actual paradox, as opposed to the idea that violating linear time is a paradox, when time really isn't linear at all. It's a fully formed 'blob' that's been there since (to have a generous cushion) a few seconds after the big bang. You might as well ask Jon to get the fuck up off the pool and go blow the head off baby Keane in his cradle instead of the 7K in the front yard. It is literally the same thing.
