Help a hopless case upgrade graphics

Short version:

My PC currently has an ATI Radeon HD 5670 card. I want a better one, but don't know what to get.

Very long version:

It was doing more or less fine until I decided I wanted to play Dragon Age Inquisition. (Heh, I bought it when it was released, it just took me this long to get into it.) Anyway, it's not up to the task, even on fairly low game settings. Today, it actually overheated (I assume) to the point my monitor stopped working and my computer shut itself off. That was a bit scary.

I'd like to upgrade, preferably before my computer burns itself to a cinder, and while I don't want to spend millions of dollars, something in the $75-150 range, or maybe even a little more if there's a really good reason, is in the budget. Problem is, there are SO MANY choices, and brands, and numbers, and abbreviations, and I don't know what any of them mean.

My greatest concern is that I'll buy one, tear open my PC, and the new one won't work, or fit in the PC, or be compatible, or be materially better than the old one. (This sort of thing happens to me enough that it's not an unfounded worry. I'll probably also find a way to destroy the old one while I'm dinking around in there, leaving me SOL.)

Please help. What's important and what's not? I assume that in general, higher numbers are better than lower ones, but beyond that, this really isn't my area. Do I need to avoid certain cards depending on my other PC specs? (I can follow up with any that are relevant.) Are there cards that are TOO good for my not-new PC? Do I focus on more RAM? Memory type? What is a memory bus and which sort do I want? I think I need PCI-express, right?

Whatever I get doesn't need to be the best in the world, necessarily, as long as it can handle a higher-graphics game without frying my computer or starting the house on fire or something. But I'm also not opposed to spending more (within reason) if it's really going to be worth it. I don't want to get the cheapest compatible card if I'm just going to have to go through all this again when a new game comes out, you know?

Like I said, I'm awfully hopeless at this stuff. If any of you have suggestions for what to get/avoid, or can clear up my confusion about what all these specifications mean...or even just point me to something that explains all this stuff using words I understand (i.e. small ones), I shall be most grateful.

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Let's check the list.

Post was last edited on October 26, 2016 4:49 PM PDT

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You are a wonderful person

Oh, lovely. I was hoping someone would say "just get this one, it's compatible and you'll like it."

So to follow up:

1. DVI-I or Dual-DVI? DVI-I is cheaper on Amazon. Do I care about Dual-DVI? (I'm guessing no, since I've made it this far in life without having to care, but you never know.)

2. Superclocked: yes or no? A quick google suggests overclocked cards stay cooler? That sounds important, but we've already established that I'm naive.

(And yes, I looked up the 960. Yikes.)

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At this price range.

We'll stick to HDMI and 60Hz links. Same for clocking. I've yet to have these overheat but I don't put them in micro cases either.

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