my home network keeps constantly dropping its signal from full bars to disconnected then again full bars, specifically on my iPad and iPhone devices.
Thanks for the screenshot.. without even asking.. that helps no end.
You can check the link. Click on the Office Airport Extreme. Hover your mouse pointer over the connection Unknown... A popup will appear with info about the link quality.
This tells you a lot of info. In my example below it shows the link is on 5ghz ie a/n. The speed at 60Mb/s might seem ok but it is going to be a problem because signal level is low.. RSSI is -78dBm .. you want that signal to be -60dB at least. (Remember negative numbers .. bigger is less.. closer to 0 is more).

Does the dropout happen as you walk around or even if seated in one spot?
If the former you can expect that to happen due to Apple not being successful with handoff in roaming. If you happen to have a Mac laptop try walking around with that and see if it also suffers the same problem. The iOS products seem to be more prone to the problem cf Mac OS.
If this is second case where you are getting droputs even sitting in one spot, then my guess is using extend wireless.
The weak link in the chain is from AC model Extreme which is wireless extended by older Extreme.
What model is the old Extreme? Check the base and give us the number, A1xxx.
eg A Gen5 is A1408 .. Gen4 is A1355.
Get rid of the wireless extend.. at least as a trial by running ethernet. Just get a long cable and experiment just running the cable on the floor. If the problem goes away you have found the problem. Get ethernet fitted properly in wall. You will need to reconfigure the Office Airport Extreme to Create a wireless network instead of Extend a wireless network before you plug that cable in, otherwise it will stop the network dead.
At one time I used a similar setup between two Gen5 Extreme. It was not a success. More like a stopgap measure.
It could also be the old or even the new extreme are old and becoming faulty. So if the ethernet does not fix it .. sorry but it is time to invest in new modern Mesh wireless or decent grade AP with ethernet links.