Broad-based public policy institutes: a great annotated listing is available at the Congressional Research Institute for Social Work and Policy
Recommended search for state-level public policy institutes/groups, etc. (note that these are often attached to a research university--though not always): search for terms such as "[state]" and "policy" and "institute" or "[state]" and "policy" and "research center." In searching for these, you may also come across many other issue-focused centers etc (which may exhibit some ideological biases). Finally, note that some states, dependent on size will have multiple centers.
For Statistical Handbooks & Almanacs, please use the following search (most of these are now available online): "[state] and statistical abstract." Note that these are also usually published or maintained by centers and institutes as well. Some of these are also connected to state data centers.