EDIT Sorry - first mistook "transitive" for "transient". Here is an example of a simple random walk on a group. The idea, indeed, is that there is a small subset of the group near the identity with a "high concentration" of dead ends, whereas elsewhere the distance from the origin is a martingale. I show that the "anomalous" behavior happens for all (sufficiently big) odd $n$, which involves some sort of periodicity (still, one can't expect all $n$ to be anomalous).
Let $G=\mathbb Z_3\times\mathbb Z_3\times\mathbb Z$ endowed with the generating set $S$ which consists of 8 elements $(\pm 1,\pm 1,\pm 1)$. For an element $g=(g_1,g_2,g_3)\in G$ let $a=a(g)$ be the number of non-zero entries among $g_1,g_2$, and let $b=b(g)=|g_3|$. One can easily see that the length of $g$ only depends on the values $(a,b)$, and the length function can be easily found explicitly by drawing a picture:
$$
\begin{aligned}
l(0,0)&=0, \\
l(2,1)&=1, \\
l(1,0)=l(2,0)&=2, \\
l(0,1)=l(1,1)&=3,
\end{aligned}
$$
and $l(a,n)=n$ whenever $n\ge 2$.
Further, one can equally easily see that the image, under the projection $g\mapsto (a,b)$, of the simple random walk $(G,S)$ issued from the identity of $G$ is a product of the simple random walk on $B=\mathbb Z_+$ with reflection at 0 and of the random walk on $A=\{0,1,2\}$ with the following transition probabilities:
$$
\begin{aligned}
&p(0,2)=1 , \\
&p(1,1)=1/2, \; p(1,2)=1/2 \\
&p(2,0)=1/4, \; p(2,1)=1/2, \; p(2,2)=1/4
\end{aligned}
$$
This observation together with the above formulas for the length function allows one to find the expectations
$$
f(g) = f(a,b) = \mathbb E_g [ l(g_1) - l(g) ]
$$
of length increments after one step of the simple random walk $(G,S)$. I will only need the values
$$
f(0,1) = -1, f(1,1)=-1, f(2,1)=3/4
$$
and the fact that $f(a,n)=0$ whenever $n\ge 3$.
Now, the stationary distribution of the above chain on $A$ is $(1/9,4/9,4/9)$, so that the expectation of the function $f(\cdot,1)$ with respect to this distribution is strictly negative. Since at odd moments of time the quotient chain on $B$ only charges the odd points, one can conclude that for all sufficiently big odd $n$ (actually, probably for all odd $n$, but I have already done too many explicit computations here)
$$
\mathbb E\, l(g_{n+1}) < \mathbb E\, l(g_n) \;.
$$
