Could just request the absolute value. This worked fine for me:
WolframAlpha["abs of fourier transform Dawson(sqrt(t)theta(t))"]
It even showed a plot.
The second issue probably comes down to having a parameter that W|A balks on. Using specific numbers seems to work fine. The Wolfram Language can handle the general parameter case using the Assumptions option.
FourierTransform[DawsonF[Sqrt[at]]HeavisideTheta[t], t, w,
Assumptions -> a > 0]
(* Out[132]= (Sqrt[a] (I w + Abs[w]))/(4 (a - I w) Abs[w]^(3/2)) *)
General remark: Posting specific examples makes it easier for others to do diagnoses and provide relevant responses. This post had such an example and that of course is quite helpful.
