As mentioned, electrons are by far the dominant charge carriers; atomic motion is dominated by thermal energy, that's all.
Resistance does increase suddenly on melting, though by how much, or with what tempco once molten, isn't very oft-studied. I think you can find a few tables of data somewhere, but it's not common knowledge. (And no, not like it's controversial or anything. It's just almost no one needs to know it.)
So, you'd expect the resistance to be high due to the chaotic environment versus a nice smooth crystalline matrix, and that's true, but the temperature is high in general so there's plenty of electron-phonon interaction and such to dissipate kinetic energy and contribute all that resistance.
Lorentz forces are negligible for signal levels as you'd have on a PCB, not to mention solder joints are so small compared to the better conductors (mostly copper or brass, sometimes steel) that are usually smooshed together -- most PCB joints are lap joints and the solder fills the mere capillary space between/around them.
At higher induction levels, depending on size and frequency, Lorentz or convection forces will tend to dominate. Lorentz force is higher at low frequencies (more flux density for a given power dissipation), while convection is dominant at larger dimensional scales (like a bath of molten metal in a furnace). With enough field intensity, and a divergent geometry, you can levitate metal in space; depending on resistivity, it will get quite hot in the process though, so that for example a copper sphere might levitate for tens of seconds, maybe minutes, in air, but eventually get too hot (resistivity too high) to support, at least at the same field intensity; beyond which it might get yellow or white hot to continue supporting it. On the other hand, if you're doing something like melting refractory metals in vacuum, let alone something truly exotic like plutonium... the high temperature is quite suitable.
Tim