Problem plotting Dawson function?

Hello together!

Plot[Re[DawsonF[Sqrt[128] x + I]], {x, 1`27 + Sqrt[1/15], 2}]

works nicely, but

Plot[Re[DawsonF[Sqrt[128] x + I]], {x, 1`28 + Sqrt[1/15], 2}]

never returns.
Do you have an idea what could be going on?
I’d be very grateful for a hint, since high precision is sometimes need in plots of plasma dispersion functions.
I have attached a notebook with some further tests, which make it look very strange.

All the best,

Klaus

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