Is my stylus damaged ?

cheaper stylii are mounted in a lump of solder and commonly fail by popping loose from the solder... this is one cause of "sliding across a record". A common magnifier will show a crater vs a needle when looking at the stylus. If the needle is there then you have an arm force problem, rebalance the arm and set tracking and anti skate force to 2.5 gms. With stylii always set tracking force at the upper end of the range unless you have an oscope and test record to verify how low you can set pressure before mistracking kicks in causing needle and record damage

If you are local to NC I service all kinds of vintage turntables to the original and new functionality.

I suspect MOFI is correct. If it still slides to the center, make sure nothing mechanical is causing it, then new stylus.

Tracking force on that Shure cartridge with the M35X stylus should be between 1.5 and 3 grams. I'd set it at around 2.5 grams.

The stylus is "sliding" across the record because you probably don't have enough tracking force (VTF) applied to the arm.

Here are the instructions to do that. It's a different model, but the procedure is the same. This is a BASIC way to do it. Using a scale is the best, but this will get you close.

I'm pretty sure your stylus is fine as long as it doesn't appear to be bent. Should be centered.

If it looks OK w a magnifier that it probably is. since your budget is tight, I would give it a try one you get the unit back from repair.