The smaller the company, the more hands-on you will be. But hands-on will vary with your responsibilities.
My first job with BSME, I was half of the department that fixed injection molds when they broke, and modified them when requested. I was 95% hands on for the first three years and did 100% of our paperwork, then our growth really kicked in and I began to hire additional people.
After five years total, I was down to ~30% hands on in the shop, milling and grinding metal pieces, setting up and running the EDM, taking molds apart and putting them back together. After a promotion, I then had two bosses instead of one, and five guys working on the floor all day long. I was still the lucky one on call at night and weekends, anytime I was dumb enough to answer my phone.
Changed jobs into the Fortune 500, I almost never touch a mill, lathe or grinder, much less a CNC machine; I don't even program them any more. I've worked for two Fortune 100 companies in various manufacturing support roles, the most I ever did was run process development trials on a molding machine. But I do have talented Toolmakers working for me who do just what I used to do but without the paperwork or meetings. I did design work then and now, working backwards from finished part design to various mold pieces to make the finished plastic parts.
Some Mfg. Engineers I know have been in Molding (rather than Tooling where I started before swinging through Molding and R&D then back to Tooling) and only operate the molding machines, robots, water chillers, etc., none of which requires much machining.
So what is your (desired) job / industry? So much of what you will do will depend on that, then on your specific company and your particular job function for that company. "Whatcha gonna, whatcha gonna do?" Or in this case, "whatcha wanna, whatcha wanna do?"
P.S.--From the company's point if view, Machinists are inexpensive, Toolmakers are expensive p, and Engineers cost real money. They aren't gonna pay you real money to do a lesser job. But with your Eng. degree, you will have flexibility and mobility within the company that the other two won't. What do you want?