Broken Charger, and now battery problems?

I have the Inspiron 15 7000. It's maybe 2 or 3 years old. Two weeks ago I went to pack up my computer and noticed that the end of the charging cable had been damaged. I'm unsure what happened, but there is a clear plastic bit on the end where the charger plugs into the laptop which lights up blue when in use. The plastic was busted and one of the wires was almost completely severed. Another wire was nicked. 

I use my laptop every day, and I noticed that when the cord was bent in just the right way, it still lit up and the laptop seemed to register that it was plugged in. So I kept using it bent this way for perhaps a week and a half while the new charger arrived in the mail. During that time, I started getting alerts that my battery could no longer power my system and needed to be replaced. The computer would also cut off with no warning. I assumed it was the broken power cord causing this problem. The battery worked normally before it broke. 

I've had my new cord for two days now. I'm concerned for two reasons. Firstly, the computer is still cutting off and displaying the same message. But second, now the fans are running extra loud and fast all the time. 

Do you think I just have awful luck and my charger and battery have bit the dust at the same time? Or could I have damaged something by using the broken charger? I'm slightly worried I may have even purchased the wrong charger somehow because of the fan issue. I doubt it but I will include specs for each, in case any of you would be kind enough to double-check me? 

This is the old (original) charger

And this is the new one