In the west marches sandbox, there was a meta-goal set up; you where exploring the territory to expand civilization and bring back loot.

The PCs where all part of an organization that did this, and there was a collective map (carved into a table).

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So, maybe having the PCs have some allegiance to some organization with in-universe goals as part of the character building, and offering support from them, might help?

I see you did have the "didn't care about" problem however. The old "you like people you help more than you like people who help you" maybe. (People convert "I helped them" to "I must like them").

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Too many rumors is an interesting problem. They also appeared to be very cheap. Maybe of rumors "cost" more? Then players won't accumulate as many.

The lack of clock means that there is no universal currency. Also, as noted, without gritty rests there your unit of adventure is the single encounter.

I'd be tempted for gritty rests, and calibrate travel based on it. An uneventful trip between close "systems" takes a long rest's time (a week?). Short cuts can take less time.

And yes, I think you need a clock. Having your ship have even abstract "supplies" stat that ticks down every day or week. To make it less of a hard deadline:

Scrap: Not enough food or repairs on gear. All ability checks have a -2. Ship has a higher chance of malfunction. Uses 1 supply point per day.
Tramp: Baseline. Enough food (but it sucks), materials to jury rig most parts, fuel that is good enough to work. Uses 4 supply points per day.
Naval: Non-expired rations, clean fuel, spare parts. Uses 10 supply points per day. After 1 month at this level, d20 checks have +1.
Lux: Non-printed food, enhanced fuel, OEM parts still in original packaging. Uses 25 supply points per day. After 1 month at this level, d20 checks have +2.

Limit how much the ship can hold in supply points.

As what you need to resupply can vary, the price of buying supply points also varies. Sometimes you need a phase coil inhibitor, other times you need to replace the baseline strain if your food paste bioreactor. Both might be 50 supply points, but the price could vary by orders of magnitude depending on where you need to buy it and what broke down.

This lets you keep the ship hungry. ;) Or not if you don't want to.

Getting a base where you can pick up cheap, reliable supply points becomes awesome.

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Once there is a supply point clock, there can be other clocks. Events can tick tock.

And if the PCs are connected to organizations that have goals (or are opposed to organizations that have goals), that provides some proxy goals for the PCs. "Explore this area" in the west marches was a goal provided by the PCs organization in west marches, as an example.

An initial task for the party to do "you have been paid to carry chocolate ants to X, then bring an ore sample back, by day Y" also reduces analysis paralysis. The rumors no on that path can be put off for later. With a bit of fudge time, they can maybe hit an other spot along the route, but only one. So there is a constraint.

I've seen this in sandbox-ish video games. Starting off with a somewhat vague "quest" that gives you something to do. Then collecting more things to do as you do that first quest. Just to keep things moving at the start. In SC2, it was "get fuel for the station". It was a vague exploration quest, and as you did it you ran into more quests, and could even complete some. When you finally refueled the base, you'd have a bunch of stuff on your todo list, and maybe 1 or 2 solid leads.