I, too, saved before each choice, and I can confirm that you are able to then play through the mission and choose the other option to make the achievement pop.
After you complete the game, unfortunately, you cannot replay all of the missions. There's no Theater or Newspaper option in Saints Row: The Third, like there was in the first two games, unfortunately.
The last mission you have a choice is the mission Three Way, where you
need to choose whether to save Shaundi or kill Killbane.
Whichever option you decide to do, you'll get a notification at the end of the mission that you've finished the game, and you can replay the final mission to see the outcome of the other choice (in fact, you can only go see the other outcome; you can't replay the option you chose first). However, this is the only time ingame where you'll be able to replay the same mission.
If you accidentally save over your saves before the other missions, your only option is to replay through the storyline. Remember though, there's still the "Gender Equality" achievement, which requires you to Play for at least 2 hours as a male character AND 2 hours as a female character. What you can do is play through the game again on the opposite gender from your first playthrough to get the achievement for this. If you just go straight through the storyline, it shouldn't take you very long.
Best Answer
I don't believe there's a completely legit way to do this.
However...
If you're willing to risk a VAC ban (however unlikely it may be), look at the Steam Achievement Manager. Sketchy-looking site, people use the tool to cheat, use at your own risk, etc. etc. etc. Here are some old Steam forums of people talking about SAM, as well as a previous StackExchange post on the topic. It's against the rules, it's a moral grey area, but for your described purposes, I personally don't see an issue with it.
Otherwise, create a new Steam account, spend $5 on something so you can download all the free games, and start fresh