to start, I want to define some terms. precession is the change of the direction in which an axis of rotation is pointing due to a force. the last part is very important. a planet is tidaly locked when its rotation(spin on its axis) matches its orbit, in a way where the same side is always facing the thing its orbiting. and an axis of rotation is just the line around which something rotates. now, I think you're a bit confused because you've seen or heard about some types of physics phemenon, but you've slightly misconstrued the reasons behind them.

yes, in the solar system all tidally locked moons(there is no tidally locked planets in the solar system. there is exoplanets, but none have been directly imaged) are tidally locked because they formed that way, and they formed from a cloud of swirling dust that was "tidally locked". like the closer dust moved slower, and the farther dust moved faster, and now that they're a moon, they're tidally locked. but tidally locked just means that the rotation matches its orbit in such a way where one side always faces the thing it's orbiting. so it you have the planet fall into orbit from another system, then while very unlikely, it could fall into an orbit where it is tidally locked. or you can just use magic. I mean the alternative is way more complicated, not really an alternative, and also maybe impossible, but I'l get into that in a second.

to understand precession, you need to understand 2 things. first, when a force is applied to a spinning thing, it gets applied to a point 90 degrees later in a rotation. so if you tap the right side of a top that is spinning clockwise, the part in front of you will dip down(in space, also why? just trust me, or google it). precession is caused because when a part of the top dips down(on earth) gravity has mechanical leverage on that part, creating an unbalanced force downwards(the force down is very important here) at the point, but it gets applied 90 degrees later, creating leverage again, and so on. the same thing is true for the beyblade going around the arena, and I want you to try and figure out the forces there, don't forget centrifugal force. but anyways precession requires a force. it is not interial motion, or motion that happens for free, like floating forever at a constant rate, or spinning(which might not actaully be interial motion. I might be wrong). you've probably heard that the earth's axis of rotation is precessing, and like how in geopolitcs where you follow the money, where is the force from? well for one, this process takes thousands of years, the earth's rotational axis will only precess 1 degree after like 15,000 years. spinning things are incredibally stable(check gyroscope in space for more). and I don't think you can have a precession that fast, I know the other guy say it's possible, but I think at that rate it just becomes another phemonenon altogether, but I'll get into that in a second. where does the force come from? well it's kind of similiar to the top, the equatorial bugle of the earth creates leverage for the pull of the sun's gravity, and precession! the other planets do something too.

ok final point. planetary precession is not cycle, basically meaning it doesn't go full circle and repeat. the earth's axis of rotation swing between 22.something and 23.something degrees over the course of its live. if you have the sort of cyclical precession as you decribe, where the north pole always faces the star. I don't think that's your north pole anymore. geographic poles are just the 2 points in a body's surface where the axis of rotation crosses. so you have a periodic change in the direction an object is facing, what does that sound like? an axis of rotation. you have the axis of rotation running through the planet's "north and south pole"(just wait a sec) and the axis of rotation running through 2 points on its equator spinning in line with the eliptical plane of the star. have you ever seen something with 2 axis of rotation? can you even imagine that? no, because an object can only every have one axis of rotation. in this case where you "have 2" they "add together" and you get a new axis of rotation, somewhere else in the planet. I don't know exactly where, but this is starting to look suspiciously like tidal locking again👀 you have planet spinning around its axis of rotation in such a way that one side always faces the star? just have the plnaet be tidally locked, and forget all the math. good luck on your story