Where are you placing the Electrum application? Are you placing it within the Applications directory? If you aren't, try placing it there and see if it stays.
Yes It's in the app directory. The icon is on the dock and there is another icon in the toolbar that opens the window.
I'll try again but I've done everything right that I can think of. I'll report back
So it is not in the applications folder and I can't figure out how to get it there. This is what I see. (shoot can't see how to insert a photo) but you know when you want to install an application it has the app which you slide into the application folder. There is no application folder. I'm not sure where to look.
I'm going to try opening security in system preferences. I also have this app Easy Find that tells you where things are and Electrum is not in application.