Having used movable do for a long time, I'm now in a country where fixed do is the norm. I'm looking for advantages/disadvantages of one over the other.
I've worked with both, and prefer movable do. In movable do, the solfege syllables represent something functional about the note in the context of the key: "do" is always the tonic, and "ti" is always the leading tone.
I've noticed that the countries that use fixed do tend to put more of an emphasis on developing perfect pitch in their students. That might be an advantage.
As someone who worked mostly with early music, my sense of pitch has to shift around to accommodate anything from A392 to A460, so I don't worry much about perfect pitch anyways.
Adam Neely briefly discusses fixed and movable Do in his latest Q&A video. In short, he says that the differences between the systems are small compared to the practical matter that English speakers primarily use solfège syllables for scale degree (movable Do) whereas Romance languages use them for absolute note names (fixed Do).
In comments, MorrisTheCat mentioned Fixed vs. Movable Do by Jody Nagel, which examines the two systems in depth. Nagel reaches much the same conclusion as Neely: pitches and scale degrees are both important to musicians, and which you emphasize is a matter of context rather than inherent superiority.
Nagel offers some advice for English-speaking music teachers that applies indirectly to your situation:
If American music students avoided solfeg syllables altogether, they would never encounter the slightest pedagogical problem when learning 1, 2, 3 and A, B, C. And yet, those little labels, Do, Re, Mi, have become such "musical-sounding" words that many teachers continue to desire to use them. For such teachers, please do not use the syllables for both absolute and relative pitch designators; choose one or the other of the two systems. If you choose to use fixed-Do, please also teach the use of 1, 2, 3 for naming relative pitches. If you use moveable-Do, please also teach the use of A, B, C for naming absolute pitches.
Thus, I recommend that you either stick with the system you already know, or convert fully to the system of your new peers, but not attempt to mix the two. Either use Do for degree and C for name, or use Do for name and 1 for degree, whichever is less confusing to you personally.
Very different systems that drive toward different understandings. Fixed do is essentially C, D, E, ... and can sometimes help develop a sense of perfect pitch (or at least vocal tension pitch.) Movable do is all about what the notes are doing and character... more of a tie in to tension/resolution and opens the door to an aural understanding of functional harmony.
To some extent, when in Rome... but the real answer is probably this: become proficient in both. Why not?