Just to clarify - with the new housing and screen the only issues remaining are the touch and intermittent lag? And that only the housing and screen were replaced, are you using the original charging port or did you end up swapping it?
My first instinct is towards the screen, given the touch issues and that you described it as a "cheap aliexpress screen". Can you provide any info on the details of the screen you bought - they may describe it as LCD, hard OLED, soft OLED, or some variation of that. If it was the cheapest, I'm concerned it might be LCD? LCD is the worst for an OLED phone, it can often have touch issues, and since the phone was designed for the lower power OLED type display, the backlight drains far more power than the phone was designed for.