My 3GS iPhone randomly shuts off

Question

My 3GS iPhone randomly shuts off or gives a low battery indicator even when the battery is not low. I tried going to the Apple store and they were not helpful saying to close out apps using the multi-task button. Even after doing that, my phone was continuing to shut down. First is was around 33%, then 54%, then 76%, then even at 97%. So I had the battery replaced with a brand-new battery.
Now not only does the phone randomly shut down saying a low battery, but it is not recognizing the charge on the battery. I had the phone on for six hours unplugged and it remained at 100% charge. Then I went to make a phone call on it and after two minutes the phone completely shut off. It would not accept a charge. I had it charge overnight. I tried using different cords. I tried plugging it into a USB port on a computer, using the wall charger, and a car charger. The phone kept saying the battery was dead. Then miraculously then phone started to work a day and a half later, but only for one phone call. Then phone died again and has not accepted a charge for two days now. You would think after two days, the NEW battery would be charged.
My phone is not under warranty. I do not have the funds to purchase a new iPhone at this time.I know there have been a lot of posts about this, but I have tried every concievable option out there and nothing is working. I'm frustrated. Is this an operating system error and can it be fixed? Or is the phone trashed? ARGH!!

Answer 1

So Brad, just an update...
My phone periodically works and then it doesn't. The battery percentage indictator still continues to remain at 100% charge though I know common sense would tell me that the battery percentage would be decreased if the phone sat unplugged and on all day long. When then phone does work, I try to restore it, but iTunes keeps saying that there must have been a connection failure and iTunes had to stop downloading. FYI- I have no connection issues with my Internet.
Here is the weird thing...
Sometimes the phone is still showing the dead battery page and asking me to charge the phone. So I'll plug in the phone, let it sit charging for 4 hours, 12 hours, even days at a time... and it will continue to have the battery dead screen asking me to charge the phone. So then I will unplug the phone and leave it alone. After approximately 48 hours, I'll go back to the phone and hit the Home button and miracuously the phone would turn on. It is as if the phone is confused and the battery life is being drained when it is plugged in, but then miracuously recharges itself while sitting unconnected to any cords or plugs.
This makes no sense and I cannot explain it. I have waited to respond to your initial postings because I wanted to see if I could restore the phone. In addition, I wanted to confirm I was not crazy and that the phone is actually acting opposite what it should in reference to charging vs. non-charging.
Any idea why the phone is acting backwards in regards to the battery life? Since my last post, I've only been able to actually turn on my phone 3 times, literally once a week and then it only lasts for about 12 hours before dying on me again.