monthly temperatue by state
It may take a few minutes of work this way...

But did find digging through NCDC's climate data inventory, the climatological rankings pages should help find what you're looking for if you're looking for their full state averages.

However, I think can help you a step further with addresses to go through: if you replace STATENUMBER in the address of:


  Temperature: https://www.ncdc.noaa.gov/cag/statewide/time-series/STATENUMBER/tavg/1/4/1895-2018?base_prd=true&amp;begbaseyear=1901&amp;endbaseyear=2000


Instead with each of the numbers 1 through 48, you can get the state data for each of the US contiguous states.  And Alaska is 50.  Likewise with precip:


  Precipitation: https://www.ncdc.noaa.gov/cag/statewide/time-series/STATENUMBER/pcp/1/4/1895-2018?base_prd=true&amp;firstbaseyear=1901&amp;lastbaseyear=2000


However, it looks like disappointingly Hawaii (and territories) doesn't get the same data produced, or at least available anywhere I can find on that site.  If you really want to seek that out, you might consider using Honolulu and Juan climate reports for a reasonable proxy, or the answer to the question Current Global Climate Dataset? on GSE may offer a rough avenue towards fair gridded information.

Not the easiest avenue, would love to see something better (and hopeful if you do go through this tedious method, perhaps post it to a Github repo or such so that more data may be easily available in the future to others facing the same problem.