In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N0(m,n) denote the number of odd Durfee symbols of n with odd rank m, and N0(r,m;n) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N0(r,12;n) by utilizing some identities involving Appell-Lerch sums m(x,q,z) and a universal mock theta function g(x,q). Based on these formulas, we determine the signs of N0(r,12;4n+t)−N0(s,12;4n+t) for all 0≤r,s≤6 and 0≤t≤3. In particular, we prove that N0(2,12;4n+1)=N0(4,12;4n+1). Moreover, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D20(8n+s) and D30(16n+t) are even with s∈{4,6} and t∈{1,9,11,13} which were confirmed by Wang. In this paper, we found new congruences for Dk0(n). In particular, for k=2 or 3, we give characterizations of n such that Dk0(n) is odd and prove that Dk0(n) take even values with probability 1 for n≥0.