What I'd do then is remove the power adapter and battery, press and hold the power button to release residual power. Now try it with the adapter only, and the battery only. Does it prove which one is causing the problem? If so you can replace it or cope with the problem, refine your result until you're ready to act.
Decent OEM AC power adapter and battery can be searched by PC model and bought on eBay very inexpensively. The cord may not be as heavy or part built as tough as the actual replacement part, however. I bought an AC adapter for a Compaq (crap) laptop for $9 which lasted a year before the PC died so it was worth the savings even if the cord was flimsy. For a better laptop that's newer it may be worth replacing with the manufacturer's replacement part.