I got accepted into a dry-lab program with 90% wet lab experience (10% dry lab). With a stats degree, I don't think you'll have any issues as long as you do well on the Q section of the GRE. It can help to take a few bioinformatics classes (machine learning, etc.) if they're offered at your school and at least have a bit of programming experience, whether it's informal, lab work, or a class. I actually consider having a wet-lab background a plus depending on what lab you work in because there are many dry/wet lab combos.
As for publications, the value of that is dependent on the program. Plenty of people get into top programs with solid research experience and no publications, which sounds like the case for you. Just apply to many programs to hedge your bets because even most top students won't get into everything.