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 ramifies 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 factorization $$(101833) = \mathfrak{p}_1^2 \mathfrak{p}_2\mathfrak{p}_3\mathfrak{p}_4,$$ 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.