To Clarify My Airport Express is Version 6.3.6
Just to clarify the clarification.. Your Express is NOT 6.3.6 .. that is the version of the airport utility.

It has nought to do with the version of firmware on the airports. Here is a screenshot of both.
And although I don't have an express here is the version of firmware on the old extreme which matches the Express.

Do you have cloud service setup.. let me recommend you disconnect it .. both in the Airport setup and in your Mac while you fix this up.
Reset BOTH your airports to factory and start over. Do the setup of the Extreme first.
Follow these guidelines. TC = Time Capsule. AE = Extreme or Express.
Factory reset universal
Power off the TC/AE.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the TC/AE to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
N.B. None of your files on the hard disk of the TC are deleted.. this simply clears out the router settings of the TC.
Setup the TC/AE again.
Redo the setup from the computer you mostly work from, using Airport Utility.
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg TCgen5 for basestation and TCwifi wireless name. Names less than 10 characters receive bonus points.
If the wireless is still a problem set wireless names like TC24ghz and set the 5ghz to different name.. eg.TC5ghz
2. Use all passwords that also comply with 1. but can be a bit longer. ie 8-20 characters mixed case and numbers.. no non-alphanumerics.
**3. If the TC/AE is main router you can skip this point. This is only an issue when the TC/AE is bridged.
Ensure the Airport always takes the same IP address.. you will need to do this on the main router using dhcp reservation.. or a bit more complex setup using static IP in the TC.
Ignore this point if you don't understand it .. post back if you have drop out issues.
4. Make sure IPv6 is set correctly in the computer. ie link-local only for ethernet and/or wireless depending on which you use. Auto is not good enough.

Once you complete the Extreme you can then setup the Express again.. do it in the same room as the Extreme so there is excellent wireless connection. Poor wireless can lead to the issue you have.