Instead of using FancyZones, you can try Alt Snap. It allows you to use ALT key + left click hold and drag, to move windows to zones. By default you need to hold down Shift as well, but this is configurable. The zone layout can be setup to go above the top edge, as a workaround to the "Chrome tabs not on the edge" bug.
RamonUnch/AltSnap: Maintained continuation of Stefan Sundin's AltDrag
https://github.com/RamonUnch/AltSnap
Optionally, I use FancyZones together with AltSnap, they complement each other well. FancyZones have better support for saving of zone layouts. Altsnap only allows saving ONE layout that is strictly tied to monitor resolution. You need to specify a different layout for your laptop, vs your monitor.
This is my current workflow when I switch from 1 laptop to using multiple monitors.
First, I choose the fancyzone layout with my cursor in the monitor, using the keyboard shortcut setup prior in the settings.
Right click Altsnap tray icon, create "test window". You will need the same number of test windows as you have zones. Make sure "Snap to Layout" option is first enabled in the configuration.
Move the created test windows to the zones setup by fancyzones. Make sure all the zones are filled with the test windows in the right places. Right click Altsnap tray icon again, "save test windows as snap layout". Altsnap will save the positions of all the test windows as the new zone layout.
To workaround the bug with Chrome tabs not being at the topmost edge, apply the autohotkey fix in Miro's answer above when repositioning the test windows, before saving it as a layout.
This looks like a lot of hoops to jump, but once you get used to it, it doesn't really take much time. You can also use keyboard to speed things up. Example, Right click Altsnap tray icon, and press "w" to create test window. "w" is underlined, meaning you can activate it using the keyboard letter instead of using the mouse.
If anyone has a better workflow do share.