Here's a map showing the results for all of North America. I created a smoothed contour map rather than showing all the points, because there are so many stations in the lower 48 that all the markers overlap. We see that Fairbanks temperatures do indeed have a modest negative correlation with temperatures in parts of the southwest US at this time of year, and there is a small positive correlation with the northern Plains and upper Midwest.
Below are the maps for each month of the year (click to enlarge). A surprising result is that the scale of very good correlations (R>0.9) across Alaska and western Canada is greatest in September; I would have expected this to occur in winter.
Update July 2: below are the corresponding maps for monthly precipitation. Note that I've used the rank correlation coefficient rather than the Pearson correlation coefficient, because precipitation amounts can be highly non-Gaussian in drier areas even on a monthly timescale.
Also note that there are far fewer stations reporting precipitation than temperature across the north (at least in the GHCN data set), so the continent-wide map shows large areas of missing data, and the results of the smoothing algorithm (Cressman analysis) are not realistic in Alaska. I recommend paying attention only to the very broad-scale features of the maps. But it's interesting to see that September again has a relatively good correlation of anomalies across Alaska.