TL;DR There is a reason your laptop came with a 120W charger. It actually needs it.
I'm also a little jealous
Your battery is draining while under charge because you are no longer supplying it enough power to drive both the battery charging circuitry and the rest of the system at once.
Your new PSU is probably 19V @ 4.7A = ~90W
Your old PSU is 19V @ 6.3A = ~120W
Depending on the components in your system this could well not be enough power.
Looking at your laptop
- Your high end i7 laptop processor is an i7-4700MQ rated at 47W.
- Chipsets (processors surrounding the CPU) are approximately 5W.
- You have a dedicated GPU, an NVIDIA GeForce GT 750M, in the laptop that is rated at 50W. This is quite hungry
- Hard drives are usually of the order of 10W of power.
- The screen is also probably between 2W and 8W of power.
Add all this together and under heavy use the components of you laptop could potentially be drawing up to 120W under full load. The charging circuitry also will be wanting a minimum amount of power to charge the batteries efficiently.
If you are only browsing the internet then both the CPU and graphics card will move to a low power state and be drawing much less. The graphics card will move to a low power mode and draw < 10W, probably similar for the CPU. The occasional burst when loading a page will not really affect the charging of the battery by much.
Under very light load your system is probably drawing around 40-50W and so the battery can charge at the same time, under heavy graphics and CPU load it could easily be over 100W, hence your 90W power brick is not going to cut it.