An adventure I quite liked, Strangers in Ramshorn used a model of standard short rests, but week long long rests and recharged magic items on a specific day at the end of the week. The structure of the adventure takes advantage of the longer downtime in two ways, it gives the PCs ample time to get attached to the NPCs and life in the town they're defending, Ramshorn, and allows all of the external threats to the town to escalate. There's essentially a set of different dungeons/encounters scattered around outside the town that all correspond to a different threat. Each week of rest that a threat isn't accounted for, it does something to the town. Spiders kill all off the local wildlife, goblins interrupt the trade caravans and make it impossible to get potions, the bandits count down to a massive raid on the town, and so on.
Significantly, the module doesn't really stop the PCs from resting at any point, as long as they can get back to town, they can take a long rest whenever they want. Instead, knowing that all these threats are poised to destroy the town and seeing how things get worse over time, it encourages them to push themselves as hard as they are comfortable with before recovering resources. Short rests are pretty much available between all encounters, but the pressure on hit dice made running out of HP a real issue over the course of the week. Much like 4e, that setup allowed the encounters to be built with an assumption the PCs were going in with full hitpoints most of the time.
The module does recommend giving slot based casters a free wand of magic missile to stretch out their resources a little, and has pretty generous access to magic items, particularly potions and later wands at a lower level than might be typical to fill in the gaps somewhat, but my players all felt it provides a pretty reasonable tension throughout.
If I was going to combine that structure with A5E's journey rules, I might replace the stationary town with something like a caravan. That gives you a big pool of Supply, presumably feeding a lot more people than just the PCs, that you can deplete and makes it a more meaningful set of journey HP, with consequences for not keeping it up.
