And if I get a new one, is there a way to transfer my files over?

Yeah - you can always transfer files as long as your HDD isn't broken. Or are you saying that you can't even turn on your laptop? I'm sure there can be some way - I wouldn't worry too much about that. You can always take your HDD out if your laptop is powerless and use an HDD connector to bring files onto a new laptop.

As for if you should get a new one - that's up to you and how attached you are to your laptop. I've been in the same place as you - I had an HP Pavilion dv7. I feel you, brother. The problems I've had with it included the fan breaking, the HDD failing, the fan breaking again, the arrow keys breaking off, the battery acting like it's not charged and randomly shutting down without warning, and finally - the death of my dear HP Pavilion - the graphics card blown out.

Even with all this, I cried when my HP Pavilion had finally blown out. Replacing the graphics card would probably cost as much as my laptop did when I got it (~$1,000) considering my computer is four years old, not even considering how difficult it would be to find a working graphics card and motherboard for an old laptop.
For me, the cost was no longer worth it. I decided to get a new laptop. I'm still really attached to my HP Pavilion and keep trying to fix (it to no avail.)

Considering you're mostly using your laptop for English papers and writing, I would most likely get a new one and stop putting up with really expensive battery replacements and fixes, especially if you're no longer under warranty. I'm sure your HP will just continue having issues down the line until it finally dies like mine did.

Getting a new laptop will be like a mega upgrade! Plus you'll be under a warranty again!
But ultimately, it depends on your attachment to the familiarity of your old laptop.