$\bullet$ Have a look at Examples and Problems in Mathematical Statistics by Shelemyahu Zacks. It is a revamp of a previous book by the same author Parametric Statistical Inference
Basic Theory and Modern Approaches.
What I like about the book is it invests proper time in rigorously elaborating every minor aspect of the concept in concern and lays out the measure theoretic language in an accessible and lucid manner. Mainly it contains some really good examples and ample exercises most of which the author has has provided solutions or sufficient hints.
$\bullet$ Another book I would say would be apt after reading C&B is Theoretical Statistics: Topics for a Core Course by Robert W. Keener. It is less measure theoretic but it is a moderate level book that covers most of the topics in decision theory, classical and Bayesian statistics that a typical graduate student should be acquainted with. While it has no vast collection of examples or problems, the latter are pretty much decent and the author does provide solutions to most of them.
I prefer a rigorous, formal, measure-theoretic development but in an accessible tone without being too concise (in that note, if you want, you can check Mathematical Statistics by Jun Shao, but I am not too much enthusiastic or inclined on using this book as a first course of study pertinent to the style I mentioned; nevertheless, there are decent collections of problems, all of which are solved by the author in a separate book) - the two books above cater to my purpose. The problems aren't exhaustive and in no way can one claim that solving them would make you proficient. But those would certainly help, for sure.
Before delving into them, please note that you do have taken a paper on measure theory.
$\bullet$ Another book that I forgot to mention (courtesy AdamO) is Thomas S. Ferguson's Mathematical Statistics: A Decision Theoretic Approach. This is a decision-theoretic book built heavily on the previous books on decision theory by Wald; Blackwell, Girshick. It is an intermediate level book with little to no measure theory. However, Ferguson is known for his plain straightforward writing style and is replete with standard examples. Selected exercises are solved by the author and he still maintains it in his homepage.
The one that AdamO mentions is another book by the same author on large sample theory, which also contains decent examples and problems, but personally I felt the book is bit condensed from the theory point of view.