If you get the same pattern of hot wires for either breaker, then the two circuits are joined in a junction box somewhere (or worse, buried in a wall.) You’ll need to go junction box hunting. They can be in curious locations.
I assume you have an AC voltage detector. If you do any wiring at all, you need one.
I’ve always separated them and I routinely come across boxes next to each other in other houses. It keeps thing neat and avoids problems like this. Assumptions can’t be made. One box one circuit. It may be legal and at times necessary due to space but it’s not preferred.
Some of the options I get an immediate intuitive sense for, but others not so much. I’d love to see schematics rather than drawings; they’re a lot easier to understand. I may bother to draw them myself, just to shed some light on the subject.
Regardless of how they’re wired, following my method above will show whether the problem is in the 3-way wiring or in an external junction box. If the power always appears at the same locations (based on switch settings) regardless of which circuit is on, the problem is outside the 3-way wiring. You can optimize this with more understanding of how the 3-way is wired.
For example, if only one power cable attaches to one of the switches, it’s a tail-end, and you can ignore it. Well, sorta …
Some things I’ve learned about the difference between people who have an electronics background and practical workers who don’t (electricians and auto mechanics)…
Most electricians and mechanics can’t read a schematic to save their lives. They like point-to-point diagrams, if not pictorial representations of circuit boxes, connectors, etc.
DIYer’s are even worse and need photographic images.
DIYer’s are afraid of wiring. Very afraid. This is not an entirely bad thing.
Most mechanics are terrified of electrical systems, which is just plain weird. I’ve seen guys who would climb under a running dragster to reach up past glowing exhaust pipes and in between running pulley sheaves to tweak something… but treat a taillight circuit like something from the Necronomicon.
a person should check all wires in a junction box, with a noncontact voltage tester, before touching bare wire or terminals. this will alert you to multiple circuits in a box or mislabeled circuits.