How to print a two color part with only one extruder
What you ask may be easier than you think. The slicing software is unimportant, generally speaking. Consider especially that your requirements fit the solution spot on.

Take a look at this thingiverse snowman.



Each of the colors is a separate model. Each of the colors is only one layer thick (0.200 mm). Each of the colors is placed with zero overlap to the adjacent colors, as the extrusion multiplier will allow enough "squish" to cause a bond to its neighbor.

You would create a model with the necessary shape for your lettering and insert the appropriate color for them. Print that model but leave it on the bed. If the bed is heated, all the better, but don't let it cool to the point of release. If it's not heated, the adhesive should hold it until project completion.

For the main background color, your model would be normal but with a single layer subtracted at the base. The subtraction would have to be the shape of the lettering. Consider that you're creating indented or recessed lettering. Use appropriate modeling techniques to perform this action.

Keep in mind that you'll want the lettering to be mirrored, as you're printing it "upside down" on the build plate.

Swap out the filament to the main color and print.

You are color limited only by your determination and patience, as you have to swap out filament and also have to create appropriate models with recesses to take the first layer, of which you'll also have to create models.

I've created at least one of the snowman pictured and it came out quite good. The layman will not be as astonished as someone familiar with 3D modeling and printing, however. You get a better appreciation when you know what goes into the project to get this result.

EDIT to add more info, rather than fill the comments:

The above method works only because the inlay is a single layer deep. If your slicer supports changing layer thickness during a print, you could print the first layer at the max for your nozzle, say 0.3 mm then set layer 2 and upwards to the resolution you desire for the majority of your print. I use Simplify3D which supports such actions. I'm not sure of the others.

Relevant to the questions posed in the comments, your modeling software may be a factor in this process. The modeling software will create/assign a pivot point or some other name to a location of reference in the model. The slicer will use that to position the model on the bed.

Here's a concept I would use to perform this task. Start with your single layer lettering, ensuring that the thickness of the letters will match your first layer height. Save the model as an STL file. Leave the lettering on the screen in your modeling software.

Create or import the case file. Position it as you wish relative to the letters. Of course, both models will be flat on the build plane or all bets are off. Use the lettering in a boolean subtraction to remove the shapes from the phone case model.

If the above does not remove the lettering model, delete that and you have the recess. Save this model in STL.

When you load the models in the slicer, they should drop into the same location relative to each other.

When you print the letters, they will be on layer one. When you print the remaining model, the cut-out portion will be on layer one, while the rest is placed atop the two color portion. The print head will move up one layer to adjust for the extra depth.

If you require the letters to be thicker than a single layer, one of the options you have is also in your question. Create the letters in the thickness you wish, use them to create the recess in the case and print separately. In so doing, you will be counting on the printer's ability to make clean bridges. The letters may fit cleanly, may have some play or slop, or be too large to fit easily. It would behoove you to make a test piece or two, perhaps a simple slab with one or two letters to see how that concept works in the real world. I was pleasantly surprised to see how well the public library's genuine Prusa i3 MK2S performed a challenging bridging operation.

I apologize for overlooking the reference to first project. With that in mind, I would consider that this is a slightly advanced project. Your modeling software and modeling software skills will obviously come into play here and is a separate subject/question/topic for discussion.