You mentioned that the symptoms got worse after you put in the sensor. Have you swapped it back to the original to determine whether the new sensor made it worse? It's possible the sensor came to you faulty.
It's also entirely possible the sensor you were sent was a counterfeit. Have you compared the old sensor to the new one side by side? Would you mind posting a picture of them side by side? I don't intend to insult your knowledge, but FCP Euro doesn't list an OE option on their website for this part:
https://www.fcpeuro.com/Volkswagen-parts/Jetta/?utf8=✓&year=2004&m=958&e=982&keywords=crank+sensor
If you go to autozone, you can find all sorts of aftermarket options for your vehicle:
https://www.autozone.com/engine-management/crankshaft-position-sensor/volkswagen/jetta/2004
Most of them are in the range you may have paid for yours (was it around $30?). If you keep digging, though, you'll find Bosch listed as OE for about $100:
https://www.autozone.com/engine-man...sch-crankshaft-position-sensor/544255_0_30433
(this is about the price I remember it being from the dealership).
I'm not advocating shopping at autozone but the part number lets you dig some more...
Here is the "same" part for less than half:
https://www.amazon.com/Bosch-Origin...006K8ZHEA/ref=cm_cr_arp_d_product_top?ie=UTF8
and if you click on the warehouse deals you can "steal" it for $25 bucks! That's less than the OEM part listed on FCP Euro's site. Is it a deal, though?
Logos, etc. unfortunately don't indicate the genuine-ness of a part.
If I showed you a Chinese character and asked you to stamp it on a 100 widgets it wouldn't matter to you whether that character translated to, "Genuine VW" part or not--you'd just stamp it out on 100 widgets as paid to do.
The reasonable conclusion is that these can't realistically all be genuine parts...certainly not with a price ranging anywhere from 25 to 100 dollars but I bet they'd all have the requisite logos to demonstrate their authenticity. If you read some of the reviews you'll note some red flags (like cable length differing, shape/color differences, etc.)
These parts are made in countries with lax regulations and very little cultural commitment to following more developed nations' IP laws. Stamping anything and everything that will indicate genuine quality is standard operating procedure and even if the employees cared they were stamping nonsense and illegal claims on the widgets, they can't read or write the language anyway and don't even know what they're doing.
Anyway, my advice is coming across like preaching now or whatever instead of being helpful. That's why I took the time to do all that linking to demonstrate that I am genuinely taking time to research what I'm posting here so you know I'm not just armchair quarterbacking. But I know what I sound like at this point and this will be the last post in regards to using an OE crank sensor from across the desk of a dealership LOL