Integrating a bivariate distribution over a region bounded by a straight line

EDITED

As suggested by JimB I added the -y to gx[x_,Y_].

 mu1 = 0; sig1 = 0.1; mu2 = 0; sig2 = 0.05; corr = 0;jointdist = 
     MultinormalDistribution[{mu1, mu2}, {{sig1^2, 0}, {0, sig2^2}}];
    cplot = ContourPlot[{PDF[jointdist, {x, y}]}, {x, -0.15, 
        0.15}, {y, -0.05, 0.05}, Contours -> 10];
    myFxy[x_, y_] := 
      31.830988618379067` E^(1/
           2 (-x (0.` + 99.99999999999999` x) - 
            y (0.` + 399.99999999999994` y)));
    coeffNorm = {0.0485058025031513, -0.006616445600455179};

    xVec = D[myFxy[x, y], x] /. {x -> coeffNorm[[1]], y -> coeffNorm[[2]]};
    yVec = D[myFxy[x, y], y] /. {x -> coeffNorm[[1]], y -> coeffNorm[[2]]};
    gx[x_, y_] = -xVec/yVec*(x - coeffNorm[[1]]) + coeffNorm[[2]] - y
    rgplot = RegionPlot[gx[x, y] >= 0, {x, -0.15, 0.15}, {y, -0.05, 0.05}];
    Show[cplot, rgplot]

    NProbability[
     gx[x, y] >= 0 && -Infinity < x < Infinity && -Infinity < y < 
       Infinity, {x, y} \[Distributed] jointdist]

   0.307558

expr = y <= -xVec/yVec*(x - coeffNorm[[1]]) + coeffNorm[[2]];

With the default value of 0 for the option MinRecursion

NIntegrate may miss sharp peaks of integrands

Setting a larger value for this option forces a finer subdivision of the integration region:

NIntegrate[PDF[jointdist, {x, y}] Boole[expr], {x, -∞, ∞}, {y, -∞, ∞}, MinRecursion -> 5]

0.3075579043682307`

We get the same result using NProbability and NExpectation with the option Method -> {"NIntegrate", MinRecursion -> 5}:

NProbability[expr, Distributed[{x, y}  jointdist], 
 Method -> {"NIntegrate", MinRecursion -> 5}]

0.3075579043682307`

NExpectation[Boole@expr, Distributed[{x, y}  jointdist], 
 Method -> {"NIntegrate", MinRecursion -> 5}]

0.3075579043682307`

If you Rationalize the numeric inputs as suggested by JimB

{mu1, mu2, sig1, sig2, coeffNorm} = Rationalize[{mu1, mu2, sig1, sig2, coeffNorm}, 0];

and add the option WorkingPrecision ->20, NIntegrate gives

0.307557904886364251344772597476334481735556904662953985238`20.

and NProbability and NExpectation both give

0.307557904886364251344772597487160029314983328842092855138`20.