I have heard that we acquire mass when we move through the Higgs field, so when a particle moves through the Higgs field, it feels a kind of opposition from the field which we see as the mass of the object, now what happens if the objects stops moving, shouldn't mass just disappear?
Elementary particles as we know them and use them in calculations of the standard model have a fixed mass, acquired when the electroweak symmetry was spontaneously broken , back in the history of the universe.
But when the universe was much hotter so that the equilibrium thermal energy was on the order of 100 GeV, these forces may have appeared to be essentially identical - part of the same unified "electroweak" force. But since the exchange particle for the electromagnetic part is the massless photon and the exchange particles for the weak interaction are the massive W and Z particles, the symmetry was spontaneously broken when the available energy dropped below about 80 GeV and the weak and electromagnetic forces take on a distinctly different look.
It happened once and since then we have the given table of particles with fixed masses.
An "object" is not the same as an elementary particle. It is composed of elementary particles but most of its mass is due to the invariant mass of the added four vectors that compose the object. The mass induced by the Higgs field on the elementary particles composing our bodies is very small.
Here is a drawing of a proton with its quark and qluon sea:
The proton has a mass of ~938 MeV, the valence quarks (which carry most of the momentum) are of order of MeV and the gluons have zero mass. All those four vectors, added up give a four vector for the proton that has an invariant mass of ~938MeV. ( The invariant mass is the "length" of the four vector of special relativity). This addition of four vectors goes up to nuclei . For atoms the contribution of electrons is small and for large objects, ~ 10^23 particles per mole one is in the classical regime where masses of molecules add and conservation of mass holds.
So "we" do not "acquire mass as we move".
The Higgs field is a scalar that couples to vector fields for W and Z bosons and matter's fermionic spinor fields. This introduces terms similar to mass terms in equations of motion. The Higgs field's own equation of motion determines for it a nonzero vacuum amplitude, to which induced masses are proportional. If you're familiar with these terms, you'll benefit from this explanation on Wikipedia. If not, you can try this less mathematical explanation.