Besides the BLF-348 mentioned above, the other 1xAAA that comes to mind is the Astrolux A01 - good Nichia neutral white, currently $8!
Another that comes to mind is this no-name stainless 1xAA XP-G R5 . Currently listed at $14.65, $13+ with BLF coupon. Not quite in your $10 limit, but noteworthy in that it’s actually a warm white LED coming stock in a budget 1xAA light. I picked up one of these after stumbling across it on the Banggood site, and I’m a little surprised I haven’t seen it mentioned here. The light itself is pretty well made, but the modes are mediocre - it’s L-M-H, but if I had to estimate the modes by percentage of final high brightness, it’s around 30-90-100… Not especially useful; basically a 2-mode light. The switch (tail clicky) is all metal and quite deeply recessed, so it wouldn’t come on accidentally in your pocket. If someone were so motivated to drop a better driver in this guy, it’d be a pretty nice light.
Finally, another with a pleasantly warm tint is the Thrunite Ti4 (I assume the slightly cheaper Ti5 has the same emitter). At $16-20, they’re definitely above your budget, but the tint is definitely in the warm range, even though they’re advertised as NW. The only reason the Ti4 isn’t my EDC pocket light is that the exposed switch is too easy to accidentally activate in my pocket. EDIT: As per the coupon code in this thread, some Thrunites at Amazon are currently 20% off, which puts the Ti5 closer to your $10 cap.
HTH!
Subscribing, as I’m always interested in new budget NW/WW lights! 