The ordinary form of the Roman Rite placed the Catholic world in a very odd position as regards Psalm singing. Instead of being the musical height of the Mass, the Psalm between the reading became responsorial as with the liturgy of the hours - which sounds fine in theory but it hasn't worked in practice. For one thing, it meant the virtual elimination of the musical treasure of the Gregorian Psalm, which was legendarily long and elaborate in order to permit spiritual contemplation and reflection. The responsorial model ends up busying everyone in a back-and-forth sing along to a tune that is typical learned on the spot. Complicating matters further, there was no music written for the texts, and so commercially driven publishers began to crank out instant music, dashed off in a flash, which is most likely what you experience week to week. This music is the opposite of the old Psalm. It is syllabic rather than melismatic. It is stuff into musical meter rather than existing as prose-based plain song. And the melodies themselves are designed to be pre-familiar so of course that means that they are drawn from non-liturgical sources.
A wonderful workaround to this problem is Chabanel Psalms, free to download and use immediately. They are good quality and yet preserve the responsorial model. This recording uses organ. We use them in our parish without any accompaniment.