Interesting then I will disregard. It's possible it's a spliced wire or something similar and the blue doesn't go all the way through. Hopefully the rest are color matched if spliced. If you find a spot where your thermostat wires are capped with another wire, or going a box. Send me a picture of that. I'll assume the wires are unspliced for right now. Because if it's a wires issue it will become apparent soon. I'm also assuming your system was working fine before the thermostat upgrade. Let me know if this is wrong. Would also love a picture of your current configuration of wires in the furnace and behind your thermostat.

Make sure your c wire adapter is configured according to your system. Video for Amazon thermostat

Thermostat to adapter should be R-R, W-W, C-C and K-K. (Looks like you might just need to move your yellow wire to K) other end of adapter will match letter to letter. R-R, W-W, Y-Y, G-G, C-C. Please take a picture of how your adapter is wired so I can confirm.

If this is proper but still have issues you will want to grab a multimeter and start checking voltages of the wires behind your thermostat. Should be in the ballpark of 22v-28v ac (target 24vac) touch R and any other wire. It's possible you have a bad wire(s), a weak transformer, bad furnace, or a defective thermostat. We're just trying to isolate the issue. You can also wire cap R and W together at this step to try and rule out the furnace. (Should get non stop heat if R and W are touching.)

As note some manufacturers and many dumb thermostats don't need 24v and most air handlers only require 12 v to engage. So bad wires and transformers can work just fine on older setups, and only become a "problem" when a smart thermostat is installed. This is just general wear and tear and not usually an expensive issue to fix, but certainly annoying.

Let me know what you find, and good luck.

Sounds like your thermostat/base might not be working if that's your current configuration for wires. I'm not very familiar with the Amazon thermostat. (I'm just a handy man) but if it's like most thermostats it won't run heat off of a Rc connection. If your thermostat is turning on it's most likely not a voltage issue, and it seems you have proper wiring. (I would still twist red and white together just to confirm heat works, but you can wait for multimeter if you want.)

I would try factory resetting the unit, contact Amazon support, or making another post here or on r/HVACadvice about your Amazon thermostat not accepting power through R. I'm past my usefulness, and at this point I'm going to recommend thinking about getting a new thermostat.

Sorry I'm not much more help. I'm only good at getting from a broad question to a specific issue. Good luck.