I have a Compaq Mini notebook that I bought a replacement battery for. I don't recall any problems with it when it was on mains power immediately before I replaced the battery. I don't use it much now and when I do I usually have to plug it in before it will start - i.e. the battery spends most of its time discharged. But it will run for a few hours when fully charged so the replacement battery is not yet worn out.
TL:DR - I would be suspicious of the state of health of your screen or of the connections to it.
...R