Let's get some colored tape, okay? A 5-pack variety pack is $4-5.
Now, on the lamp, mark the black wire RED. Red is the preferred color for switched-hot, and presumably you want the lamp to be switched.
BLACK is preferred for always-hot. YELLOW is Harper's preferred color for travelers.
These "preferred" colors are not Code requirements, but ...
OK, so here's what we know about 3 cables
(and can guess about the 4th). Note that "color coding" is not happening here and the color codes will be a chaotic rainbow, with no consistency whatsoever. This situation is why I own 10 colors of tape.
The only consistent color rule is if neutral is present, it must be on white. And that's the end of ...
Grats on realizing the importance of the screw colors. That matters. Position doesn't.
Now, the problem is, it's clear that travelers are not properly identified. Now, I'll tell you a Harperism: I like to positively identify travelers, and when I know what they are, I mark them with yellow tape. Both travelers can be yellow, they are interchangeable and ...
That should be fine. The only question is how many neutral (white) wires are connected and if two, are they connected to separate wires to the fan. If there are two and they connect separately, you should swap them too.
As @Jack noted, make sure to turn off the breaker before working; don’t depend on the switches being off.
There are two travelers on each of the existing switches, no neutrals. If there's nothing else in the single box, that common wire goes to the fixture. The second box is where your feed comes in. Don't mix up your white traveler and connect it to a white neutral. You will probably end up with two white wires going to the switch in the second box, one ...
A 14/3 with a 3-way switch is going to be:
Either hot (in) or switched hot (out)
That means you don't have a neutral, which is a must for a receptacle (switched or not).
However, there might be a solution if you can use smart switches. Many 3-way smart switches only need one traveler. You could, potentially, install smart switches and ...
Ordinary switches don't need neutral. Many (but not all) dimmers, timers, smart switches, motion detectors, etc. need neutral. There are some exceptions that:
Leak a little current through the circuit when "off" - works fine with incandescent but often not very well with LEDs
Use ground - permitted for very low current in limited ...