Bound of an integral function

Let $$f(r):=\int_{\mathbb{R}^d}\left|\int_{\mathbb{R}^d}e^{2\pi i \langle x,y\rangle}e^{-|y|^2+r^{1/2}|y|}dy\right|dx,$$for $r\geq 0$.

Is it possible to uniformly bound $f$ on $\mathbb{R}_+$? i.e.$$\exists c>0,\forall r>0,f(r)\leq c$$if not, can we do it on intervals $r \in [0,U]$?

Tried writing Taylor expression for cosine and exponential but this seems to complicate the problem. Can we eliminate absolute value?