>>1365574Push. Carry takes the extra energy of lifting the entire weight, as opposed to letting the wheels carry the load. Exception is if it would get stuck, or it won't roll, or for very, very short, direct shortcuts.
Pic somewhat related.
>>1365588Try sticking a block undeneath your bottom bracket so your rear wheel is off the ground, or clamp your bike to a chair or something and try to replicate it. or at least see what's hitting what where when you shift. Wild guess, broken/heavily worn teeth on the big ring so it's falling off? Weirdly misaligned or bent FD?