So even I manage to find a 5v, 300mAH adaptor, it will cause unstable current which will damage the battery?
There's no such thing as a 300mAH adaptor. mAh, means milliamps per hour and is a unit of charge, not current.
Current refers to the number of electrons flowing per second.
Charge corresponds to the total number of electrons transferred, irrespective of the length of time.
Charge is analogous to the amount of water in a tank and current the amount of water flowing through a pipe, at a given instant.
A charge of 300mAh is: 100mA flowing for 3 hours, 150mA for 2 hours, 300mA for an hour, 600mA for half an hour etc.
What you have is a 5V power supply, rated to a maximum current of 300mA. This means no more than 300mA can be drawn from it, before there's a risk of it either overheating, or the output voltage dropping below. mA means milliamps, which is a unit of current, not charge.
The TP4056 module is a battery charger. It will deliver a fixed output current of 1A, into a battery, until the voltage reaches 4.2V, then it stops. As mentioned above, the current can be reduced to 100mA by changing P
PROG to 12k.