This isn’t an uncommon problem, and there are techniques for dealing with it.
I could’ve written this post, a couple years ago. I learned at session 0 that everyone wanted less combat and more intrigue, and had to make that style fit for 5e D&D. As such, everything I suggest below are techniques that I’ve discovered through long-term play of a combat-light D&D5e campaign (about 3 years running, at this point, with characters at 11th level).
“My character sheet is mostly useless!”
Remember that a class is more than the set of bonuses you get. Every class comes imbued with some flavor that gives the character hooks and knowledge outside of combat.
A fighter is a martial expert – someone with a deep understanding of tactics and strategy, and maybe one who the fellow party members look to when violence has already happened or may happen, to avoid a solution. A monk is usually part of an order – their path isn’t necessarily something they’ve discovered on their own. Tie them into an NPC group, especially a powerful or resourceful one. Warlocks create their own story hooks, via the existence of their patron, and most spellcasters have enough out-of-combat utility that they get to use a lot of their abilities on the regular. Give the rogue advantage to notice someone has a concealed knife, even if there’s no hard mechanic for it. If this isn’t enough, find reasons for the party to use their “abilities” outside of the rules. This will often require homebrew, but find something the characters are interested in, and tie it in to the game mechanicaly.
In short, bring out the flavor part of their classes and abilities, and make them matter.
Combats aren’t draining resources in a meaningful way
This is an easy trap to fall into. Once you realize you are only going to have one combat per adventuring day at best, you have to start changing the way you build fights. Throw out CR and the combat deadliness charts from the DMG (almost) entirely while using this advice.
Note that all of these solutions can cause a session or two to be very combat heavy. As long as this isn’t all the time, you’re still fulfilling your promise of lower combat, but verifying that the party is happy with the occasional extended bit of combat is a good idea.
The two major solutions I’ve used to great effect are the gauntlet, and the extended combat. both solutions somewhat hinge on time pressure to some extent. A third option provides an alternative for when you cannot provide pressure the same way. You can have all kinds of hooks for the time pressure, including ones that might make the out-of-combat intrigue easier or more interesting if the party beats a deadline. Tiered rewards based on completion time might cause the party to expend more resources faster in order to get the better “reward”. For example, the party discovers that Tim is going to destroy some evidence then perform a ritual that does “bad stuff”. If they go in guns blazing, they might be able to preserve the evidence while burning a bunch of resources, but if they go in a little safer, they might come out a bit safer, but at the cost of the evidence.
1: The Gauntlet
This is a whole adventuring day compressed down to a small period of time. The time pressure keeps the party from backing down and taking a rest. Tim the Enchanter will finish his ritual in 2 hours if the party doesn’t get through his dungeon and encounter him in his lair, and the party must fight through a small number of linked encounters in a mini-dungeon crawl to get to him, for example.
2: The Extended Combat
This is, in a sense, the same solution as the Gauntlet, but even more compressed. This could be something like a phased fight, with Tim the Enchanter standing behind a wall of force while his minions attempt to kill the party. After the party grinds through the minions, Tim summons more minions and joins the fray. After Tim takes serious damage, he transforms into his true demonic form, and fights the party one last time at “full power”, only after this is he defeated.
Do not build your combats with higher CR monsters, expecting the party’s resources to keep up. The higher tier monsters will just crush the party before they can expend enough resources to win. Instead, you want a slow release of monsters of a reasonable combat equivalence to the party. Draw out the resources over time during a longer combat.
3: The Inescapable Dungeon
My campaign world includes dungeons that the party can optionally enter that provide a reward at the end that hinges on the party completing the task in one go. These can be somewhat meta, if you aren’t careful, but the ancient magical ruins that can tell when someone has left and lock them out of rewards can provide a more traditional session or two that lets the combat-focused characters shine. These are something that got added to my world specifically to help solve the one encounter adventuring day problem, but they do require a spot in your world and a reason to exist.
Use Milestone Experience
This isn’t something you asked about, but I found it very important. Milestone experience helps you abstract advancement away from combat. With you running less combat, advancement will be unbearably slow, unless you stop tying combat to XP.