To my mind, the overriding rules here are these:
Alternate Form
The creature retains the type and subtype of its original form.
Undead Type
You cannot have an undead that has a Constitution score, so since Alternate Form has you retain your type, you cannot gain a Constitution score. I believe this is “more specific” than the general rule that you gain the target creature’s physical ability scores when using Alternate Form.
You could argue the opposite way, however, that Alternate Form is a more specific case than the general rule that undead do not have a Constitution score. But note that this results in a living (has a Constitution score) undead (retains that type), which to me is a good enough reason to side the other way.
As for using Charisma in place of Constitution, this is a common feature for undead, but not a universal one. In some places, that privilege requires a feat (e.g. Undead Meldshaping in Magic of Incarnum). Depending on your build, I might want to tax you for it; Cha-to-everything is a fairly potent strategy as it is, and undead is a desirable type anyway. Getting to use Charisma for yet more things, and avoiding the “weakness” of the undead type’s lack of Constitution, might be problematic. But if you weren’t already pursuing a Cha-to-everything build, I’d probably give it freely.
The Dominate Beast spell will end
There was a (somewhat confusing) developer tweet about this topic. Jeremy Crawford stated (emphasis mine):
In #DnD, the exceptional trumps the general. (No longer being a valid target trumps condition carryover.)
When a wildshaped Druid reverts to their normal form, we have those two conditions: the target changing type, and condition (charmed) carrying over. The target changing type makes it an invalid target which, as confirmed in the sage advice tweet, trumps the charmed condition carrying over.
Since the spell would no longer has a valid target once the druid changes back, the spell will end.
What about the other two Dominate X spells?
For completeness, let's run through the two other dominate spells.
Dominate Person requires a humanoid target. All official PC races are humanoid, so while the druid is in its normal form it can be targeted and charmed by a Dominate Person spell. If the druid wildshapes into a beast, its type changes to match that beast. So, the druid is no longer a humanoid, the target of Dominate Person is invalid, and the spell ends. In this way, it works in reverse of the Dominate Beast spell.
Dominate Monster works on any creature. Whether in beast form, humanoid form, or even in a Circle of the Moon druid's elemental form, the druid is still a creature. Thus, even going in and out of wild shape won't make the druid an invalid target. Since the target is still valid, the Dominate Monster spell will still have full effect, even if the druid wild shapes.
Best Answer
No, not unless your new shape is physically capable of producing poison.
You may, for example, be able to enter into a Barbarian's Rage, or attempt a Fighter Manuever. However, this is a physical trait, similar to darkvision, so unless the new form also has the trait, you do not benefit from it. You may have skin, and your new form may have eyes, but your skin does not become poisonous, and your eyes do not gain darkvision.