How can I get my Particle-Board Cabinet re-staining project looking more professional please?
Do what a professional would do - don't touch it when wet.

Sand the affected areas with fine grit and re-stain, leave until it dries completely, then move, flip over and do the opposite side, etc. Apply the poly in the same manner. You cannot handle pieces until they dry fully. You also cannot handle them with hands that are wet with finish.