It's hard to say without hearing the noise you describe as crackle -- it could be mechanical in nature, or it could be as you suggested, electrical distortion. An over- or under-powered amp or a poor impedance match could cause the circuit to be overdriven into distortion, and headphones vary widely in that regard -- it's entirely possible that the headphone amp in the new laptop is a poor match for those phones; on the other hand, it's possible that the new laptop has a problem. However, it sounds like you're saying this was already a known issue (if a minor one) with that particular pair of headphones and other equipment, and the new laptop displays the same issue but to a much greater degree..? In that case a defect in the laptop would seem less likely. I'd suggest a few tests if possible, to try and pinpoint the cause..
- Try a level test: is the crackling present only at louder listening levels and the signal is clean at lower levels, or is it present at any level and just varies in degree at different listening levels?
- Try other headphones (beg & borrow if need be) with the gear, particularly with the new laptop -- do any of them exhibit the same problem, or is it restricted only to the headphones in question? Ideally, you'd test with both earbuds (which are usually designed to be a good match for computer audio outs) and proper studio & hifi headphones.
- Try the problem headphones with as many pieces of gear as possible; if the crackling is present -- to any degree -- with all of them then the phones may need more repair than you were able to do;
- the cable connection(s) inside the phones could be loose, a problem that developed gradually in a couple of my headphones from frequent pulling on the internal connections;
- the headphone's diaphragm itself could be damaged (or possible rattling from excessive subsonic energy)
If the headphones only exhibit the problem with certain devices -- or certain types of devices, like computer/player audio outs, as opposed to hifi or studio mixer headphone amps -- then it might be worth it to touch base with the headphone manufacturer (if possible) to verify if the spec (i.e. impedance) should be suitable for the equipment you're using them with (look up the specs for the laptop and other devices those headphones have a problem with, as well as for those they work ok with).
If you can pinpoint the conditions that cause the crackling to occur, then you can decide whether a repair is in order or simply a different pair of phones is needed for the new laptop.