Engrave a design into slate sheets I currently have multiple sheets of slate measuring 8"x12" at 1/4" thickness, and various gauges of colored wire (picked up the wire from the Beading section of a hobby store).

I have been scaling a map of the U.S. onto freezer paper, that once complete will be a 26"x40" map outlining the individual states.

I would like to engrave the map into the slate, and then fill the channel with metal to make it more visible. In some tests, the only metal that was softer than the slate was solder wire (tin / lead, not sure of the %s) which allowed for hammering it into the channel for a good fit. With any metals I tested, they were too hard and would essentially hammer through the slate, and so I have decided to use a jewelry metal glue to inlay the wire securely. This gives me a little leniency on the depth, and width of the engraving, but I am looking to use either 18 / 20 gauge wire.

I have tested a few different Dremel bits made for engraving, but I am currently using the Dremel accessory 108 1/32" Engraving bit, with the depth adjustment attachment. I have okay results, though find it difficult to keep an accurate line. Additionally, I had a lot of issues placing a sketch overtop the slate and trying to engrave through following the pattern on the paper. I held the Dremel perpendicular to the slate, and attempted to use it like a router, but it still had a tendency to walk a bit. While browsing the Dremel site, I think it looks like the '125 1/4" Carving Bit' may better suit my needs, as it would have better cutting power in lateral movements.

Would it be better to keep trying to leave the paper on top of the slate, or to cut out the individual states and trace them with the Dremel? While I would prefer to try and keep the depth consistent, would it be better to handle the Dremel freehand, more like a pen, than perpendicular to the slate?
If you're using the rotary tool with a suitable engraving bit, you may find it easier to trace the state profiles by attaching the paper to the slate with an adhesive. Even a simple glue stick will wash off with water once the job is completed. By tracing around the shape, the lines you generate may be more consistently positioned by not being centered on the line.

Your question about handling the tool is going to be tougher to answer, as it is more dependent on your skill than on one practice being better than the other. With respect to the cutting bit used, unless it is designed to cut with the tip, the perpendicular placement may cause problems.

Another option you have would be to construct a caddy, a sort of "pen holder" to keep the tool at the best angle for engraving, providing a stable base while getting optimum cutting. A tool caddy will also enable you to accomplish a more consistent depth of cut.

If you have a maker space in your area, you may also consider to visit to see if they have a laser engraver available for use. I have a CO2 laser which will do a wonderfully accurate job of engraving slate, to nearly any reasonable depth (1-2 mm typical). You may find other members of the maker space equally helpful. I personally would be happy to take on this type of project for a fellow maker.

Directly related to the above, if you are unable to find a makerspace, consider to visit a laser related forum or two and cast about for a local laser user. If you are willing to pay shipping in both directions, your options increase. I use RDWorksLab occasionally, a forum aimed at users who own lasers with the Ruida controller line, but have since put most of my time in LightBurn, a forum for that particular piece of software. SE does not have direct communications between users, but you can find my username in either location if the shipping option works for you.