Oof, the lack of a separating comma. =P
Linguistically, there are two valid ways to interpret this clause.
- The spell damages objects in the area
- The spell ignites flammable objects that aren't being worn or carried
- damages objects in the area
- ignites flammable objects
that aren't being worn or carried.
In the former case, your interpretation is correct: Meteor Swarm would only ignite objects that are not being carried, but would damage all objects, regardless of being carried or not.
In the latter case, your interpretation is wrong: the spell would both damage and ignite objects, but only if those objects aren't being carried.
"So which is it?"
The precedent of other spells is that it's the latter. The wording of most other spells is specific:
It ignites flammable objects in the area that aren't being worn or carried.
—Fireball, Player's Handbook, pg. 241
A nonmagical object that isn't being worn or carried also takes the damage if it's in the spell's area.
—Shatter, Player's Handbook, pg. 275
The lightning ignites flammable objects in the area that aren't being worn or carried.
—Lightning Bolt, Player's Handbook, pg. 255
So it's certainly possible that the phrasing used in Meteor Swarm (which is shared with spells like Delayed Blast Fireball, PHB 230) is just a consequence of trying to smash together two clauses without considering the awkwardness of the linguistic construction.
On the other hand, that Linguistic awkwardness: the designers could have avoided some confusion by instead wording the clause like this, but instead they chose not to:
The spell damages objects, and ignites flammable objects, in the area that aren't being worn or carried.
Yet another reason commas are so valuable!
So ultimately, as DM, I bias towards the precedent established by other case studies in the game. Most other effects/spells that can cause object damage are very clear that they do not do so to objects that are carried—or if they do, they're particular about the conditions under which they do so. One could understand breaking that rule for a powerful ninth level spell like Meteor Swarm, but the fact that this wording also shows up for spells that are a lot less powerful suggests this was not the intent.
So probably, this spell is not supposed to cause damage to objects that are carried. Both because it is linguistically valid to interpret it as such, and because the precedent of other spells is to not cause damage to carried objects.
One last thought: the vast majority of objects that can be carried would be immediately destroyed by the damage of any of these spells I've quoted. If the intent were for this damage to be applied to those objects, then it would obliterate all non-magical items (and probably the magical items as well, since the damage resistance is not impregnable) within the radius. As I mentioned: that's kind of understandable for a powerful ninth level spell, but it's far less likely to be the case for a much weaker spell like Delayed Blast Fireball, which uses the exact same wording.