The second half of the Overchannel feature says:
If you use this feature again before resting, you suffer 2d12 necrotic damage for each level of the spell, […] This damage ignores resistance and immunity.
It only ignores resistance and immunity, not vulnerabilities. Although many resistances and vulnerabilities of the same type of damage usually cancel, this is not always true. This answer confirms that both would be applied if you had them.
The rules on damage resistance and vulnerabilities only mention cases where you have multiple sources of resistance. These are not superimposed and would in any case be ignored in this case. The best rule that this gives us is:
Resistance and then vulnerability are applied after all other modifiers.
So, you would first apply your resistance, which is ignored, then your vulnerability. The result is that you suffer double damage with this feature.