Are 4*AA batteries sufficient to power my servos with the below rated amps

I do not know the price of Amazon. On Aliexpress, for example, the usual price for a 18650 with a capacity of 2500-3000 mAh is about $3 for a piece. Normal manufacturers are Varicore, Litokala, LG, Samsung, Panasonic.
Never take 9900mAh batteries, this is fake.
A simple charger costs $8-10.
If you need a more powerful source, you can connect 2-3 batteries in parallel, but only of the same type.