Is it normal for the end of my strip to look dimmer than the start?

Yeah that's classic voltage drop. At 5V you don't have much headroom, so by the time the current travels 12 feet of strip traces plus whatever wire runs you've got, the far end is sitting a half volt or more below the near end. On an addressable strip that shows up exactly how you're describing: dim tail, whites going warm or yellowish because the red channel sags less than blue. And you noticed the other telltale too, it only shows at full brightness because that's when current draw peaks and the drop gets worse.

Glad someone explained the yellowing part, I never understood why my old setup did that. I had the same issue behind my TV and fixed it by feeding the far end off the same supply, no second one needed. Just make sure the supply is big enough for the whole strip length. I used some leftover 18 gauge from a doorbell project and it's been fine for a couple years now, short run like yours should be no problem. Only thing I'd double check is that you connect positive to positive and ground to ground at the far end, I mixed mine up the first try and got nothing but flicker.

One detail from when I did this behind my bookshelf, if you inject at the far end only you can end up with a reverse gradient where the tail is now brighter than the middle, so feeding both ends from the same supply junction keeps things even. I soldered to the pad points at my cut mark rather than using the clip-on connectors since those add a bit of resistance of thier own, and heat shrink over the joint so nothing can short against the cabinet hardware down the road.

I'd take that doorbell wire anecdote with a grain of salt. Behind a TV you probaly never run full white at max brightness, so any marginal wiring would hide itself. My take is 18 gauge is right on the edge for injection on a 5V run, and if the OP is seeing a visible tail dim alredy, cheaping out on the injection wire defeats the whole point. The copper in old speaker wire can also be tarnished where it's been sitting on a spool for years, which makes soldering annoying even if the resistance is fine. I'd spend the few bucks on proper stranded wire sized for the job and not wonder about it later.