Horizontal line between verses

This isn't really built into MuseScore the way you describe it but you can open Format->Style->Text->Lyrics even (or odd) and make the lyrics underlined then all lyrics of that type verse (even or odd) will be underlined but still not a continuous line.

You wrote:
I've tried experimenting with the horizontal line in the "Lines" menu but this offsets lyrics.

Try this:
Before you move the line (and mess up the lyrics), click on the line and uncheck 'Automatic placement' in the Inspector. Then move the line between the verses you wish to separate.

You could (mis)use a melisma line, starting with a Ctrl+Space and followed by as many _ as it takes

Incidentally, I have recently tried exactly the same thing - adding a horizontal line after every 3rd verse as this makes the verses lot easier to follow when changing the systems.

Both methods below involve quite a lot of manual adjustments, so preferrably apply them as the last step. Even so this may backfire when changing fonts, uploading to musescore.com or using the score with the mobile app etc.

Method 1 - lines

Pros: independently customisable line (style, color, thickness, hooks etc.)
Cons: lines need to be adjusted for each system even if they are continuing across multiple systems

Steps:

1. anchor the line to the first note of the staff and expand to the right as needed (Ctrl-Shift-Right multiple times).
2. turn off autoplacement for the line via inspector
3. set the placement to below the staff.
4. adjust Y Offset (via inspector) for verses 3-7 (some can be grouped) to make the gap between verse 3/4 and 6/7
5. Adjust Y Offset and appearance of the line - you can change independently its thickness, colour, style (solid, dashed etc.) add hooks etc.
6. The line may span across multiple staves, however its continuations need to be adjusted separately anyway (e.g. offset and auto/placement above)

Method 2 - melismas

Pros: works across all staves it has been applied to
Cons:

1. re-numbering the verses needs to be done one at a time and in special order
2. the internal numbering becomes out of order
3. the default melisma thickness is too heavy for my taste but its thickness cannot be adjusted separately; any change affects the "real" melismas as well
4. handling of melismas may change in future versions, so at some point it may stop working.
5. it is quite difficult to find the anchor of the line - neither the line nor the invisible text are clickable (you need to click the first lyrics in the verse above or below and then use down/up arrow to select it)

Steps:
(Ideally, you would add almost empty verse 8 and push it upwards to 4 by changing verse number via the inspector but at the moment it does not work that way - see my other post above and Bug? moving verses up and down via inspector.

1. start with the most bottom verse (7) and re-number it via inspector to 9
2. then re-number verses 6, 5, 4 one by one to 7, 6 and 5 respectively
3. create verses 4 and 7 with invisible lyrics with melismas (Ctrl-Space as invisible text and _ multiple times to create the melisma line)
4. adjust Y Offset (via inspector) for verses 4-9 (some can be grouped) to make the gap between verse 3 and 5 smaller
5. adjust the melisma thickness if desired (Format / Style / Lyrics).

As a variant of this method, you could leave the original verses as they were and add melismas in two new verses (8,9) and adjust their Y offsets as desired.

Another method?
Not really, I have also tried underlined lyrics and frames around them but no great joy here:

This is so far what I came up with. Maybe someone else has any other smart idea?