Unfortunately, the walker in the waste’s capstone is a notorious source of a great deal of confusion. This is a somewhat unusual case, but every walker in the waste wants to know the answer because it also affects Level Adjustment.
Exception-based rules, or “specific-trumps-general”
The d20 System is an “exception-based rules system.” The way it’s designed, there are general rules for how everything works, but then specific items break those rules in specific ways, as delineated in their text. Wizards of the Coast had and has extensive experience with such rules from Magic: the Gathering, of course, which has general rules like a creature cannot tap or attack on its first turn (unless it has haste), a creature must tap to attack (unless it has vigilance), and so on.
Within the d20 community, this is usually referred to as “specific-trumps-general,” referencing a phrase in the core errata document.
The class features of the dread necromancer and the walker in the waste are specific cases, relative to the general templating rules. Thus, they should take precedence. But the question is, how much precedence?
Dread necromancer: specifically and explicitly doesn’t work
Dread necromancer, at least, is easy:
When a dread necromancer attains
20th level, she undergoes a hideous transformation and
becomes a lich. Her type changes to undead, and she gains all
the traits of the undead (see page 317 of the Monster Manual).
She no longer has a Constitution score, all her existing Hit
Dice become d12s, and she must reroll her hit points. A dread
necromancer need not pay experience points or gold to create
A dread necromancer who is not humanoid does not gain this class feature.
Here, the specific case directly addresses this question, and everything is clear.
Walker in the waste: specifically.... not mentioned at all.
Walker in the waste, on the other hand, does not:
On reaching 10th level, you learn to apply the secrets of waste preservation to your own body, becoming a dry lich. You must undergo the Sere Rite, overseen by another dry lich, which includes preserving your flesh, removing your organs and storing them in special canopic jars, and imbuing your body with foul magic to make it undying. See the dry lich template, page 155, for more information.
As a dry lich, you cannot be permanently killed unless the canopic jars containing your life essence are destroyed.
That’s the entire text, with no mention of requisite types. This specific case says that you “[become] a dry lich,” and “must undergo the Sere Rite,” leaving no room for that to not work or be impossible, but it also specifically references the dry lich template.
Plus, there isn’t any general case to trump
Worse, achieving 10th level in the walker in the waste prestige class is the only listed way to become a dry lich. The template cannot be applied to a creature who has not already done that, which makes it bizarre for it to have general rules that the specific walker-in-the-waste case is meant to trump.
This comes up occasionally in discussion forums on the question of whether or not the walker in the waste class feature applies the Level Adjustment found on the dry lich template. It is generally assumed that “rewarded” templates like these do not apply the LA, since if they did there’d be no point to the reward: the original case, the dragon disciple, explicitly lists the effects of gaining the half-dragon template, and does not mention the LA. Since a character could just start with the half-dragon template if they were going to take on the LA +3, there’s no reason for it to be applied. Dread necromancer is a similar case for the lich template, though in a non-epic game gaining LA +4 at 20th level means nothing. But the walker in the waste is the only way to get the dry lich template. Why would it even have an LA if that were to be waived?
Ultimately, there is no satisfactory answer. The class feature’s very existence implies that it’s supposed to be a reward, supposed to be an exception, supposed to allow you to do something you generally couldn’t. It’s not much of a reward to let you do something you could do anyway. But the case of the walker in the waste make all of this very murky, since you generally cannot become a dry lich at all.
Your DM should allow you to benefit from the walker in the waste capstone. You’d already started on that path, and the prestige class is basically there for the purposes of getting the dry lich template; there’s no point to it if you don’t. To completely waste your levels in the class by making you wraith-spawn and thus barring you the template, would be an exceptionally jerk move on your DM’s part. The wraith thing seems like it was a reasonably cool and interesting plot twist, but it shouldn’t screw you over like that. If you didn’t have walker in the waste, I’d encourage the same for the dread necromancer’s lich transformation, even though the class explicitly bars it, for the same reasons.
I probably would not allow you to simultaneously be a lich and a dry lich, though. Dread necromancer’s got plenty going for it, plus you could just take a prestige class instead of dread necromancer levels towards the end if you wanted.