I have a used reproduction Remington cap lock revolver that functions well with one glaring exception. All safety notches on its cylinder - that is, machined slots between nipple seats into which the hammer nose fits - are too shallow. I place hammer's nose into a "trough" [rather than a slot]. But if I attempt to rotate the cylinder, it rotates into having a nipple beneath hammer nose. Safety notches are not.

Since slot positions are correct, merely too shallow, it is likely that altering them would be neither difficult nor time consuming if alteration can be done freehand or via "eyeballing." But I am no gunsmith. I do not have tools to do the job. And if I had the tools, confidence is lacking.

How do I solve the problem without buying a replacement cylinder?

Carry it like you do a cartridge SAA; 5 chambers loaded with the hammer sitting on a dead nipple.
Or send it to me and I will mill them deeper for you. Is your revolver a Uberti or a Pietta; they are different.