I am looking for number fields $K$ which satisfy the following properties:

1. $[K:\mathbb{Q}]=5$.
2. The Galois closure of $K$ has Galois group $S_5$.
3.  For each prime $p$ which *ramified* in $K$, there exists a prime ideal $\mathfrak{p}$ of $K$ of inertia degree and ramification index $1$ above $p$. i.e. we have
$$(p) = \mathfrak{p} \cdot \prod_{i=1}^r \mathfrak{p}_i^{e_i},$$
with $N(\mathfrak{p}) = p$.

It is well-known that one can construct infinitely many fields satisfying conditions 1. and 2. using Hilbert's irreducibility theorem. It is the last condition 3.
which is the most important one, and says something like the ramification of $K$ is very mild.

I have been able to find such fields by looking at databases of number fields. For example one such field is given by
$$t^5-t^4-5t^3+5t^2+2t-1 =0$$
This has discriminant equal to $101833$, which is prime. One checks that we have the factorisation
$$(101833) = \mathfrak{p}_1^2 \mathfrak{p}_2\mathfrak{p}_3\mathfrak{p}_4,$$
where each ideal has norm $101833$.

>> Do there exist infinitely many such fields?

I'm possibly willing to weaken condition 2., to ask instead that the Galois group is a solvable (but still non-abelian) subgroup of $S_5$, if it helps. However in my application I cannot remove condition 1.