No codes running rough

Dge magnum rt 5.7 2005, when idling car has a slight shake to it but while accelerating white smoke comes out the tail pipes. Feels like my hemi is only running on 4 cylinders. Overtime Had to replace battery three times and multiple fuses that burnt out. No codes pop up and check engine light turned off fairly quickly. Had my rubber shift coupling replaced, center bairing replaced, my water pump replaced, thermostat, an exhaust leak taken care of so I was told, new bcm (which was the cause of the breakdown). Car then sat for 9 months then taken to a shop. They said Accuater for ignition was broke? But replaced it. New key replaced and an oil change was done. Still no codes or engine light on. But still idling rough and white smoke while accelerating. Shop had suspicions of hydraulic lifters not running and mds out. But I didn't want to pay another 1600 for something that was unkown. Had a shady mechanic work on it during the time is sat for 9 months and he tried to steal my car by having it towed off. Having bad luck with trustworthy mechanics and still no clue what the real issue is.