Solving 5 nonlinear equations, is there a suitable way?

AbsoluteTiming[
 xyz = Reduce[{exp1, exp2, exp3, exp4, exp5, Element[{x, y, z}, Reals]}, {x, y, z}, {a, b}];
 ans = NSolve[{exp1, exp2, exp3, xyz}, {x, y, z, a, b}] // Quiet;
 ]
 ans
 {exp1, exp2, exp3, exp4, exp5} /. ans

Output

{0.466927, Null}
{{x -> 0.222444, y -> -2.15701, z -> -0.686049, a -> -0.200401, b -> 2.10858},
{x -> 0.155142, y -> 0.904622, z -> 0.950293, a -> -0.0124473, b -> 0.489938},
{x -> -1.95192, y -> -0.545867, z -> 0.119973, a -> 0.00314125, b -> -0.0762384},
{x -> 1.13873, y -> 1.76806, z -> -0.573138, a -> 0.317141, b -> 1.86267}}

{{True, True, True, True, True}, {True, True, True, True, True},
{True, True, True, True, True}, {True, True, True, True, True}}

Introduce a sixth variable for the exponential term

    eq6 = E^(x - z) == exz
    {eq1, eq2, eq3, eq4, eq5} ={exp1, exp2, exp3, exp4, exp5} /. E^(x - z) -> exz

And tell NSolve all variables are Reals

    (nsol = NSolve[And @@ {eq1, eq2, eq3, eq4, eq5, eq6} && 
             Element[{x, y, z, a, b, exz}, Reals], {x, y, z, a, b, exz}]) // Timing


    {0.641, {{x -> -1.95192, y -> -0.545867, z -> 0.119973, 
              a -> 0.00314125, b -> -0.0762384, exz -> 0.125947}, 
    {x -> 0.155142,y -> 0.904622, z -> 0.950293, a -> -0.0124473, b -> 0.489938, 
     exz -> 0.451513}, 
    {x -> 0.222444, y -> -2.15701, z -> -0.686049, a -> -0.200401, b -> 2.10858, 
     exz -> 2.48058}, 
    {x -> 1.13873, y -> 1.76806, z -> -0.573138, a -> 0.317141, b -> 1.86267, 
     exz -> 5.5393}}}

    {eq1, eq2, eq3, eq4, eq5, eq6} /. nsol

    {{True, True, True, True, True, True}, {True, True, True, True, True, True}, 
    {True, True, True, True, True, True}, {True, True, True, True, True, True}}

Even Solve is very fast.

    (sol = Solve[And @@ {eq1, eq2, eq3, eq4, eq5, eq6} && 
          Element[{x, y, z, a, b, exz}, Reals], {x, y, z, a, b, exz}]); // Timing

    {0.797, Null}

Try

exp1 = y*E^(x - z) == 18 a*x + b*y;
exp2 = 18 a*x + b*y == y*(8 a*y + b (x + z));
exp3 = 0 ==  18 a*x + 72 a*z +2 b*y;
exp4 = 9 x^2 + 4 y^2 + 36 z^2 == 36;
exp5 = x*y + y*z == 1;
Reduce[{exp1, exp2, exp3, exp4, exp5}, {x, y, z, a, b}]

which adds some of your equations to others to eliminate some of the exponentials AND it removes the Reals argument which I have seen greatly slows things down at times.

Perhaps you can extract the Real solutions once you see the results.

Check all this very carefully to make certain it is still correct.