This is the common problem of Samsung battery connector. The cables don't make good contact with pins of battery connector on mainboard.
Here is example of good connection
The second one is example of bad contact
Don't assume your old battery is bad because new one has the same problem of loose contact.
The easy way to fix is using Silver conductive glue but you have to use with care. Wipe off any extra glue between terminals with alcohol, otherwise you will have a shorted circuit with symptoms like overheated, low or high battery temperature.
This is example, no glue between cables
This is a sample from Ebay
https://rover.ebay.com/rover/1/711-...85&hash=item4d622308f1%3Ag%3AzFQAAOSwyXNZ5XVn
One red ( positive ) or black ( negative ) cable lost contact, you never get full charge or battery capacity drops quickly. Most users always blame the battery is bad due to the quick discharge, but the real problem is from loose battery cables.
Another possible root cause is the cracked solder joints of battery connector on mainboard.
This one needs to be resoldered and this one is 99% found on all samsung tablets above 10 inches
This battery connector was fixed, more solders added to fix cracked solder joints. These cracks you won't see unless using microscope, without experiences, most will see it looks normal.
This rework is beyond the scope of most users, you may need the help from a repair shop
Most Tab 3 need to fix both battery connectors. After that, it can last another 1 or 2 years.