Engrave a design into slate sheets
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.