I just talked to Sony repair service and they told me to send it in for replacement, would take up to 2 weeks and I have to pay for shipping as well. Would cost me around 10⬠as I also need to buy some shipping material.
Hence I opened up the gamepad to check if it's really the spring that broke and it was. The spring is completely gone, though, which I find weird as it wasn't flying around inside the controller and there was no rattling when I shook it.
I looked for some DualSense springs online and thankfully found them in stock at some console/gamepad customization shop, 10pcs for 3,99⬠and expected delivery tomorrow or friday.
I find the situation with these springs breaking much more worrisome than stick drift issues.