The possibility of changing the cofinality of a regular cardinal without collapsing any cardinals is equiconsistent with a measurable cardinal:

On one hand, if $\kappa$ is measurable, then by Prikry Forcing one obtains a model in which $\kappa$ is singular of cofinality $\omega$ and all cardinals are preserved.

On the other hand, work by Dodd and Jensen shows that if $\kappa$ is regular but becomes singular in some generic extension where all cardinals are preserved, there is an inner model where $\kappa$ is measurable.

EDIT: $K$ is the so-called core model which is an inner model similar to the constructible universe $L$. Notably, $K$ is not changed by set forcing (this has been stated in Consistency strength of some problems about singular cardinals which answers the same question but i do not know an exact reference). So suppose $\kappa$ is regular in the model $V$ and singular in some set forcing extension $V[G]$. Then in particular $\kappa$ is regular in $K^{V[G]}$ (the core model constructed in $V[G]$) which equals $K^V$ by the preceding remark. As $K^V\subseteq V$, $\kappa$ is regular in $K$, hence measurable.