The help comment only refers to the much deeper randomness engine as it may be used in effects. Otherwise randomness is always re-seeded per layer for properties that natively access these functions. It's a compromise in how this has to be as simple enough for average Joe who would expect this or else effectively nothing would be different and you'd have to provide extra controls. AE takes the reverse approach. You would have to pin it with the seedRandom() function in an expression selector and then write your code around it, e.g. a normal wiggle or noise function. I'm wondering, though, whether you could not do everything in one layer? You could create the fat stroke as a layer style, you know and use the text stroke for your flickering secondary stuff...
Thank you very much for the insight!