How to get dynamic shadows without using a lot of stationary/moving lights?

How can I get dynamic shadows without using a lot of stationary lights, they cause a big performance drop.

Do I need to change a certain amount of them to stationary or just place some stationary lights near the static ones in particular areas to create shadows?

My game currently has almost no shadows on moving objects and it looks kind of weird but I can’t change all lights to stationary due to the performance drop.

You can’t expect having many shadow-casting lights and good performance at the same time. You can have as many stationary lights as you need, but their shadow-casting performance will depend on number of objects/triangles they are affecting.

To put it straight, you should try to design the level in such way, that no more than 2-3 shadow-casting lights are visible at any given point in the level and number of dynamic objects, affect by those lights is kept to a minimum. If you are working with mobile, the constraints here are even tougher.

So I basically need to fake it with static lighting and then just put a few stationary lights so that the shadows still look good?

Pretty much yes.