In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $\mathbb{T}^d$, in which case $1\leq p\leq 2$, and $\mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $d\leq 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least $d/(d-4)$. This contrasts with the usual $L_p-$theory where, independently of the dimension, adaptation requires regularities to lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation, centred at adaptive estimators. Those are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend more globally to weak losses such as Sobolev norm distances with negative smoothness indices.