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I'd be more than happy to assist you. I'm going to go out on a limb and make the assumption that you have some technology experience. I say that because you have a great router, and you're interested in the Echo dot, if my assumption is incorrect please let me know and I will break down my instructions into greater detail.
Your ASUS router supports 2.4GHz and 5GHz. I suspect you're connecting to the 2.4GHz frequency using the Echo which typically is the safe route. However I suggest trying to switch the Dots over to the 5GHz frequency and see if you have better luck.
Is your wireless being broadcast as one SSID for both 2.4 and 5GHZ? Or do you have something like...
If so you want the 5. If not, we can setup your router to broadcast the two frequencies separately to achieve this. Then I would connect the dots to the 5GHz frequency and see if that resolves your issue.