Help with difference between two means

#1
Dear statisticians,

I have four groups of values, A, B, C and D. I'd like to find out if the difference between groups A and B is significantly different to the difference between groups C and D.

For example:

Group A: [30, 21, 22, 26]; mean 24.75; stdev 4.11
Group B: [11, 16, 13, 12]; mean 13.00; stdev 2.16
Group C: [38, 39, 40, 40]; mean 39.25; stdev 0.96
Group D: [24, 24, 20, 23]; mean 22.75; stdev 1.89

I started by calculating the difference between two means (A-B and then C-D), then the mean squared error, and then the standard error of the difference [sqrt(2*MSE/4)], followed by the confidence interval (SEDiff*t) where t for a two-tailed distribution with 6 degrees of freedom is 2.447. This gave me the following results:

Group A-B: meanDiff: 11.75; MSE: 10.79; SEDiff: 2.32; 95% CI: 5.68
Group C-D: meanDiff: 16.50; MSE: 2.25; SEDiff: 1.06; 95% CI: 2.60

Because the 95% confidence intervals overlap (11.75 ± 5.68 vs 16.5 ± 2.6), I take this to mean that the difference between the differences isn't significant. Is this correct, or am I miscalculating/missing something? Also, if this is correct, how can I calculate p for the difference between A-B vs C-D?

Thanks in advance for any advice!