Your difficulty in 3-times firing will be getting the lower temperature glaze to adhere to a pot that is already fired to clay body maturity, ^10, and is not absorbent anymore. Can be done, but its not easy nor reliable/consistent. Plus, reheating the glaze already on the pot could/will impact that glaze, also.
You can get bright colors at ^10 if you fire in oxidation. Some underglazes will hold their color to those temperatures. Others will not; it will be a matter or testing to see what work on your claybody and in your kiln.
The tumblers look as if the iron oxide was applied at greenware stage and bisque and then glazed in white (with a wax resist for the oxide highlight), followed by an oxide wash and/or underglaze brushed over the white glaze. If using underglazes on top of a glaze, you'll want underglazes that are less refractory and more likely to melt into the glaze for coloring. That could be one alternative to try. There are likely others, too. The fun is in solving the puzzle before you.
As the clay dries and shrinks around the toothpick even before firing, it could still crack. Plus, the wood will combust during firing early, leaving the spike unsupported as the work is fired to higher temperatures.
We need a few more details . . . Is the glaze flaking off the bisque ware or after being glazed fired? Is this a problem with all glazes, or just a particular glaze? Are the glazes newly mixed? Or one's that have been around for a while? With a variety of forms or just one form? And, if you could post a picture, that would truly help provide an answer.
There was no need to remove the glaze from the cups; just wash them to remove any oils, dirt. Use Sharpies with oil-based ink or Porcelain Pens designed for writing on glazed wares, then bake in the oven following your directions. Exterior use only. Google "Sharpie pen drawing on plates" and you'll get a trove of DYI links that explains the process.