Here are two observations. Both struck me as surprising at first and then not so surprising at all. I don't know that either helps with the asymptotics.

- $a(m,n)$ is the expected number of flips of a fair coin until one gets either $m$ heads or $n$ tails. 

- Consider the recurrence relation $$b_{m,n}=b(m-1,n)+b(m,n-1)+2^{m+n-2}$$ with $b_{0,n}=b_{m,0}=0.$ Then $$a_{m,n}=\frac{b_{m,n}}{2^{m+n-2}}.$$

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I suppose the first observation translates to the behavior of right/up walks in a rectangle. The second observation easily results from the recurrence for the $a_{m,n}$

Here is a table of the first few values $b_{m,n}:$

 $$\left[ \begin {array}{ccccccccc} 1&3&7&15&31&63&127&255&511
\\ 3&10&25&56&119&246&501&1012&2035
\\ 7&25&66&154&337&711&1468&2992&6051
\\ 15&56&154&372&837&1804&3784&7800&15899
\\ 31&119&337&837&1930&4246&9054&18902&38897
\\ 63&246&711&1804&4246&9516&20618&43616&90705
\\ 127&501&1468&3784&9054&20618&45332&97140&204229
\\ 255&1012&2992&7800&18902&43616&97140&210664&
447661\\ 511&2035&6051&15899&38897&90705&204229&
447661&960858\end {array} \right]$$

One observes (and then easily proves) that 

-  The table reduced $\bmod 2$ is a Sierpinski triangle.
- $b_{1,n}=2^n-1.$
- $b_{2,n}=2^{n+1}-n-2.
$