Usually, this would be down to a poor earth or corrosion on bulb holders, but since you have retried the filament ones, this seems less likely - unless the new LEDs actually draw more current than the filament bulbs, which would be unusual (possible if they are very bright though).
I'd try connecting only one of the LEDs at a time, and retesting by turning on the headlights. It might tell us something.
Many LEDs have a diode bridge incorporated, so they can work in either polarity, but others do not and only work one way round. Trying with each LED unit in turn, switch polarity, retest with headlights, and let us know the effect (if any).
Lastly, LED units can generate electrical noise, which can upset other electronics - some brands are worse than others. You may be better off returning them for a refund and buying a well-known brand (++£) if possible, from a big, established website-based supplier - eBay is not always a good source for Chinese electronics....