To make the result of NSum more precise you can use also the NSumTerms option (15 by default, see e.g. Numerical Evaluation of Sums and Products) appropriately increased. Let's try e.g. :
a1 = NSum[ HarmonicNumber[2 m]/m^3, {m, 1, Infinity}, WorkingPrecision -> 140, PrecisionGoal -> 70, NSumTerms -> 2000] 1.9746275368413284954203787248027995910222173561519748313727983181550691548
now compare
NIntegrate[ x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 75, PrecisionGoal -> 70 ] N[ Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a1, 75] % - %% 0.0778219793722938643380943991911599389199168078241333818284167516820632583615
0.07782197937229386433809439919115993891991680782413338182841675168206325836 0.*10^-75
That's pretty close.
Regarding your definition of b I had no problems with evaluating it, e.g.
N[b, 70] 1.974627536841328495420378724802799591022217356151974831372798318155069
You can see that the NSum result is really close to this value.
For some closely related problems with NSum see e.g. this question Precision differencesPrecision differences.
