The head should not 'walk down' under load if you have tightened the z-axis lock lever. Having the fine feed engaged mostly stops any drop in the head and tightening the lock absolutely prevents it. As above, the cutting forces can pull a cutter right out of the R-8 chuck, especially an import collet like most of us start with. An endmill holder and Weldon type mills prevent that . One of my better investments was a 3/8 and a 1/2 inch holder. The spring kit is nice, but it will not fix the pull-out problem.
You do not have anti-backlash nuts on the unmodified version of the mill. The nuts can be slit and turned into a mostly useful adjustable nut, but that is not where the problem is in most reasonably well maintained mills of that type. Much of the backlash is usually in the thrust bearing preload. The nuts that hold the hand wheels on also set the preload. Getting the load set and the locking nut tight is tricky. The remaining backlash is in the nuts and you can probably live with it. I never had more than 0.010 of backlash, which is easy to compensate for in a manual machine. Not so with CNC, which likes minimal backlash for accuracy.
The move to CNC for the mill should not be undertaken until you have made some small projects to plan. It is not a cure all for a sloppy mill. I converted mine. I did it in 2 stages. The first stage used the screws, nuts, and factory dovetails. It worked and I was able to produce a pistol receiver to plan from it. The stock screws had barely tolerable backlash, the acme screws sucked op torque, and the dovetails sucked. Stage 2 lengthened the Y-axis, re-cut and scraped the dovetails, installed ball screws, added rotation to the head, and converted to belt drive. Before you head off down the CNC road, there are some considerations. The mechanical parts in kit form cost as much as the mill did new. Import and home brew versions of the mechanicals are about half that. You trade shop time and aggravation for cost. The electronics to drive the machine will be in the neighborhood of twice the cost of the mill. Again, you can save money with low end import boards, but may have to buy replacement boards in short order. The software to run the machine will be at least the cost of the mill. All told, I would recommend that you set aside $2K for the project. That's a lot of money, but not as much as the $4K factory CNC from LMS. On top of that, if you make your own mechanicals instead of a bolt on kit, You will need a lathe and a second mill, preferable a larger one. I did mine with a G759 (G704 w/dro) It would have been easier if I had access to a knee mill.