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I've noticed after watching a lot of anime that sometimes "Watashi wa" sounds
like "Wataxi wa" or maybe "Watakshi wa." The instances I've heard the latter
were always from females. I was wondering if there's a different accent,
dialect, or linguistic custom behind this pronunciation? Thank you.

For example, I heard it at 4 minutes 20 seconds into the video here:

<http://gogoanime.io/basilisk-episode-14>

(I used the top link). It also happens again at about 4m 38s.
CC BY-SA 3.0
2016-05-27T12:45:44.223
34422
2018-07-02T05:34:05.283
14560
question
english-to-japanese
Watashi versus Wataxi, are they different accents?
I guess you are hearing わたくし watakushi, which is like わたし watashi in a humbler
manner. Nothing special about it.
CC BY-SA 3.0
2016-05-27T13:16:23.853
34425
2016-05-27T13:16:23.853
1065
34422
answer
Actually Watashi わたし and Watakushi わたくし aren't only for female. Male also use
them, especially in serious situation like business or public speech, etc.

You can often hear Watakushi when a politician gives a speech.
CC BY-SA 4.0
2018-07-02T05:34:05.283
59833
2018-07-02T05:34:05.283
30470
34422
answer