The menus, again, work in that the little menu windows open up and the selections can be clicked, but clearly there's something basically wrong with the layout. The pop-down menus have no spacing around the edges, much as the main menu entries lack spacing.
I'm running the xfce4 desktop in an otherwise plain Ubuntu (i.e., not Xubuntu) installation, which is what I've been doing for many years now and which works fine on the older (16.04) laptop. I suspect that what's wrong (or missing) is some GTK setting or configuration, but I have not been able to figure out what's right on the old machine but wrong on the new one.
This is a dev build (3184) of Sublime 3, for what that's worth. Other applications seem to work without any weird menu layout problems.
edit — I've tried uninstalling the
apt version and using
snap as suggested in a comment, but that gets me nowhere. The
snap install seems to work, and something ends up in
/snap/sublime-text, but there is no apparent way to run it. In fact nothing I've read about how the whole
snap facility is supposed to work has any apparent bearing on what actually happens on my system.
more info — Switching to a Unity session makes Sublime (and the few other things that were affected, like
lxappearance) work normally. Thus I think the problem has to do with something about my xfce4 setup, so I'll look into that.