Can I use something similar to "Slide Vertices" to create new vertices and edges? I use Slide Vertices (hotkey Shift-V) to slide multiple vertices along a face. I also use the tool in edge mode to slide edges along a face. Does Blender have a tool that does the same function but leaves the original vertices/edges and also creates new vertices/edges that slide along existing faces?
Here is an example workflow to clarify what I mean:

I select multiple vertices that make up an edge or edges (call these original_verts)
Use some tool/shortcut that I hope exists in Blender
Use mouse to slide the selected edges along adjacent faces
Release mouse, where new vertices/edges will be created on where I released, but original_verts remain in their original positions.
Here is a related question: Duplicate and slide edge (rip from boundary)
I don't think there is a way to do exactly what you want in vanilla blender. But there are a few things you can try depending on what you want to achieve exactly, and what kind of mesh you're working on.

Just for a quick reference, instead of sliding verts with Shift-V, you can do the same more quickly by just pressing GG while your vertices or edges are selected.

The first thing that comes to mind is the Offset Edge Slide tool. Not to be confused with normal Edge Slide. All you have to do is select the vertices you want, and then press CTRL + SHIFT + R. This will slide your new vertices where you want, but as a downside it will do so symmetrically. So you will have to clean up the side that you don't want by just dissolving.


Here's an example video:



Another option that could work how you describe is just a simple extrude. The key is to change the snapping options to edge, and when you are extruding hold CTRL and hover over the desired edge. Once you've completed the extrusion, you will need to merge duplicate vertices by pressing M and selecting &quot;By Distance&quot; since this method will create duplicate geometry.

Example video:


Hope this helps.