This is actually a pretty common way to run RPG sessions.
Most of my experience running RPGs (Including D&D) in the past several years has been in 2-4 hour convention slots. The campaign I'm currently running is also structured as a series of self-contained episodes with three-hour time limits. There are a number of techniques you can use to pull this off, without having to actively run a timer or resort to heavy railroading:
Make sure everybody knows the time limit
In my current campaign, we all agreed up front that sessions would last three hours, no more and no less. This means that everyone has a clear idea of when the session is drawing to a close, and conscientious players tend to work together to make sure they keep the story moving, especially towards the end of our time. Because it's a friendly group and not a con slot, we sometimes go 30 minutes to an hour over, but for the most part we try to stick to the three hour time limit. It's a lot easier to do this when the whole group is working together, as opposed to me trying to do all the time management myself.
Use combat sparingly
Even the shortest combat scene in D&D takes disproportionately more time than any other kind of scene. In my three-hour sessions, I generally count on being able to run at most two combats, and usually only one. When I've run four-hour D&D sessions for fundraising events, the most combat we've ever been able to do in those sessions has also been two encounters, so I'm pretty confident that that's the limit. I try not to run a combat unless it represents a decisive point in the story -- random encounters are a no-go for this kind of play. It sounds like you're already leaning towards a low-combat campaign, so this should be no issue for you. Also, remember that monsters can run away or surrender if a combat seems to be going against them, and this is a great way to save time.
Try to limit yourself to 4 "scenes" per session
For several years I've participated in a yearly charity event where we run a four-hour session of 5e with a strict time limit. The quests we run are designed in a modular format where players pick from a "quest menu" to decide which challenges their table wants to take on. The quests are divided into three categories: Combat, Puzzle/Exploration, and Roleplay. Combat obviously takes the longest, Exploration the second-longest, and Roleplay generally goes the quickest. The goal for each game table is to complete 5 quests from the quest menu in an evening, and my tables have always managed to do this, albeit sometimes by hurrying through the last quest or two.
Five "scenes" in four hours means an average of forty-five minutes per scene. That means for a three-hour session you can probably run four scenes if you keep up a brisk pace, and three scenes if you're taking your time with them. If you're not doing any combat, you might be able to go up to 5 scenes, but I'd be wary of trying to cram in any more than that unless your players tend to really blow through social encounters.
Think of scenes as story beats instead of encounters
When we plan Encounters as DMs, the tendency is to rigidly prepare a specific situation. Obviously this is necessary for a good combat, but for more social scenes it can lead to an inflexible adventure structure that may make you feel pressured to railroad. When planning your sessions, think more about goals, and less about exactly how your players will achieve them. Do the players need to gather a key piece of information? Defuse a tense standoff? Foil a villain's plot? Each of those goals can be accomplished in a scene, but you don't necessarily have to know what that scene is until your players tell you how they want to achieve the goal. Stat out key NPCs and make maps of important locations, but don't get too attached to exactly how your players will interact with those elements. This will allow you to keep your sessions on schedule without resorting to really strict railroading.
Plan multiple successive end-points for a session
If you want players to have freedom to concoct their own strategies and go at their own pace, you're never going to be able to fully determine how far they'll get in a session. They might skip over a key encounter and surge ahead, or they might spend a full 90 minutes discussing a genius stratagem and fall behind. Rather than forcing players to keep a predetermined pace, I like to go into every session with at least two possible endpoints in mind: one that represents what I see as the ideal endpoint of the "episode" and another one that represents a "cliffhanger" moment about halfway through the "episode." That way, if the players go slowly (which, in my experience, they almost always do) you can still give them a satisfying ending, and just use the other half of your planned adventure for the next play session. Your players will have no idea they're "behind schedule," and you'll buy yourself some extra prep time too!
As an example, I recently ran an episodic session that was meant to end with the players climactically facing off against a small-time crime boss in their home town. The party spent more time than expected concocting an elaborate scheme to gain audience with the crime boss and pump him for information, and by the time they finally initiated combat, it was time to end for the night. So, as a way of ending the session, I described the crime boss's angry reaction to their deception, narrated his minions charging towards the party with weapons out, and then told them we'd pick back up next time. The players couldn't wait for the next session, and spent the interim plotting combat tactics for their inevitable showdown. The next time we played, we jumped straight into the combat, and we were able to take our time running a good fight and narrating a fitting close to the adventure. I never mentioned to the players that they'd gone slower than I'd expected, and no one was the wiser.
If you want, you can also plan a further endpoint for if the players move too quickly. But in my experience, players moving too quickly has never really been a problem. It's much easier to slow players down (especially ones who like to roleplay) than it is to speed them up, so plan accordingly.
TL;DR: Plan on running 3-4 scenes per session, with at most 1-2 focused on combat. Tell your players the time limit, and put a "cliffhanger" endpoint about halfway through your planned session in case the players are slow.
