My four year old iPhone 4 will not turn on? Replaced battery, still unresponsive?

Verizon CDMA iPhone 4

Both sides are shattered, but the phone has been in working condition.

In the past two weeks, my phone has been having battery problems.

It started with just randomly turning off, but it wouldn't come on at all.

Recently it had drained down to ten percent, and rapidly declined.

I left it plugged up overnight, and it was maxing out at 65% battery.

After that, it would not take a charge at all. It slowly declined down to two percent, and died completely.

It was constantly booting and staying on for short amounts of time on the charger. I formatted the phone.

I got it replaced three days after (today), and it came with 56% charge.

It started to turn itself off again, and the battery started to drain with each reboot.

It eventually drained down to 19%, and then died again.

I've tried to start it by holding down the power and home button; but with no response.

iTunes does not detect it when I plug it up, and I'm running out of ideas.

I'm going back to the shop tomorrow, but I need to know if the battery is the problem.

It doesn't seem to fix my problem.

Any help would be well appreciated, as I want to avoid buying a new phone at the moment.