Piggybacking off this some more (refreshing my memory of the space program in the 60s after binging For All Mankind), suppose some version of the Manned Orbiting Laboratory is put into orbit.
Just spitballing here: the MOL (and maybe some Gemini Bs) is built prior to cancellation, but not launched. So it gets transferred to NASA and they decide to throw it up into orbit as a purely civilian space station - no point in letting the hardware go to was. Some modifications are made so it can dock with an Apollo command module, and NASA decides that its nice to have the extra flexibility that the station offers. They run some experiments, maybe send up supplies ahead of one of the lunar missions, basically just get the hang of operating a more complex and modular range of missions. For bragging rights, the US has now beaten the Soviets to launching a space station, as well as landing on the moon.
From here, maybe the MOL module can be connected with Skylab, to provide a larger overall station, and build up more experience connecting different modules together. This could be an excellent opportunity for the International Skylab mission to be put into effect.
A few knock-on effects: first off, it would be real nice if Robert Lawrence didn’t die, so he can be the first black astronaut in space (maybe even on the moon in this scenario). An interesting possibility is that, if the MOL is there, followed by Skylab or Skylab B, for long enough, maybe NASA decides to pull the trigger on the Big Gemini spacecraft, which makes supplying the station much easier than with Apollo - more volume. However, if NASA does go with Big Gemini, that could delay the Space Shuttle for quite awhile. That may turn out to be good for the Shuttle program - a later Shuttle could be better designed and maybe even approach the original goals of the program.