You do not apply advantage separately to the effects of a single saving throw, but you do not get advantage in this case
Most spells require only a single saving throw. This is explicit from the way the save is described, as in dissonant whispers, which says:
The target must make a Wisdom saving throw.
This is singular - there’s only one save, even though there are multiple effects. Spells are explicit about the effects of failing the save; in the case of dissonant whispers:
On a failed save, [the target] takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you.
While many spells (mostly cantrips) leave it at that, which tells us that nothing happens on successful save, dissonant whispers is also explicit about that case:
On a successful save, the target takes half as much damage and doesn’t have to move away.
So both effects are covered by the single saving throw: you either take full damage and flee if able, or take half damage and don’t have to flee. Having advantage or disadvantage, however you get it, would apply to the saving throw as with any other d20 roll - you roll twice and choose the result you want. It does not split the saving throw into separate parts to which advantage or disadvantage may or may not apply.
There are a small number of spells with separate saving throws for separate effects, but where this is the case each saving throw is mentioned explicitly in the spell’s description.
In the specific case of a halfling getting advantage on the save for dissonant whispers, though, you’re out of luck. The halfling trait Brave says:
You have advantage on saving throws against being frightened.
Frightened here refers to the condition detailed in Appendix A of the Player’s Handbook and Basic Rules. Spells and effects will use phrasing like “on a failed save, the target is frightened” to make it clear when it is being used. Dissonant whispers does not mention being frightened, and so Brave does not apply. (Narratively, the movement is framed as a result of the “discordant melody” causing “terrible pain”, not fear.)
Of note is that no spell combines damage with inflicting the frightened condition in a single effect. Effects that cause a target to become frightened may specify additional effects that apply while the target is frightened (for example fear, which forces a character to flee each turn they remain frightened, or weird, which inflicts a separate save against damage on the frightened target at the end of each of its turns), or that happen at the moment the character becomes frightened (fear also causes a creature who fails their save to drop whatever they are holding). But if failing a save would cause the frightened condition, whether or not there are other effects, then a halfling gets advantage on the saving throw.
So: you roll a single saving throw, without advantage, and either take full damage and are forced to flee, or take half damage and aren’t. You do not save separately against the damage and the other effect, for this spell or any other, whether you have advantage or not, unless the spell specifically says so.
If your DM wishes to split saving throws so there are always separate ones for damage and additional effects, whether for the purposes of advantage applying or in general, that’s a big change to the standard rules. It will, among other things, slow down play and greatly affect the efficacy of spells. I’d recommend talking to them about the reasons for this change.