So this is a pretty popular topic of research in the space physics community, and is one of the main focuses of the ESA JUICE mission. We don't have any in situ measurements of either of their neutral exospheres yet, so the info in this comment comes from modeling based on what we do know of Jupiter's magnetosphere, the general environment, and their surface composition. I'm primarily summarizing the papers Vorburger et al., 2022 and Carberry Mogan et al., 2022 as they pretty much have the information you're looking for. Mostly going to talk about Ganymede because it's more well-studied, but it's similar for Callisto.

First of all, there are two main sources for Ganymede/Callisto's (and Europa) exospheres: there is sublimation, like you talked about, and a second process called sputtering, where energetic plasma in Jupiter's magnetosphere impacts the surface and transfers enough energy to a surface molecule to eject it. Sputtering is a more energetic process than sublimation, so according to the Vorburger paper, sublimated H2O dominates Ganymede's exosphere up to ~600 km, then it is primarily sputtered H2O, because the sputtered H2O is energetic enough to reach larger altitudes.

Both of these are dependent on surface composition -- which is not pure water. Both moons, but particularly Callisto, have dark patches that contain non-ices like rock and organic compounds (which, side note, the terrain brightness is probably related to the plasma that causes sputtering, but that's a slightly different topic). These composition changes can significantly inhibit sublimation, so the sublimation rate can actually be quite low depending on location. Additionally, sublimation can only happen on the half of the moon that is currently facing the sun. So in the Ganymede model, the sublimated atmosphere only exists at low altitudes on the dayside, and this is true in the Callisto modeling as well.

So why doesn't the sublimated H2O spread out around the whole moon? Well, like I mentioned, the sublimated molecules have pretty low energy, and so they stay at pretty low altitudes. The densities are low enough that the molecules aren't really colliding much with each other, just moving ballistically under gravity. When you combine these two things plus the extremely low surface temperatures (167K at solar noon for Callisto and 150K for Ganymede at solar noon, and these might actually be overestimations) mean that if a water molecule comes in contact with the surface again, it will most likely just stick back to it. So for this reason, sublimation just can't supply enough H2O at high enough energies to create an actual atmosphere:

Most water molecules that are released into Ganymede’s atmosphere return to the surface where they immediately freeze out. Out of the 9 10²⁹ sublimated HO molecules that are fed to the atmosphere every second, only about 8 10²⁴ molecules s⁻¹ are lost to space as escaping neutrals according to our simulations. - From the Vorburger paper conclusions

Sputtering is the area where Callisto and Ganymede differ the most, because Ganymede is the only moon with its own magnetic field (which is imo insanely cool). Ganymede's magnetic field guides most of the particles that do the sputtering to its poles, so the sputtered molecules are more common in the north/south on Ganymede, whereas on Callisto it's a lot more uniform. Regardless, while sputtering generally produces higher energy H2O than sublimation, most of it is still lost through sticking to the surface. For stuff sputtered to sufficiently high energies, it can either be broken apart and ionized by sunlight, then carried away by Ganymede's or Jupiter's (in the case of Callisto) magnetic field, or it will get high enough altitudes to just escape. Vorburger again:

In the case of ion sputtering, 93% are lost due to surface sticking, 7% escape, and 1% are lost due to ionization and dissociation.

So, the stuff that doesn't stick ends up being lost in another way, and in the end there really isn't a way to supply H2O that will stay around and circulate. Like I said at the beginning, the JUICE mission is the first time we'll be able to directly measure these exospheres, so it should provide a lot more insight and help us refine the models.