I agree that taking a bio course or two would be helpful to demonstrate your commitment. If you can find a research position that would dramatically improve your chances, in my opinion. Since you have a background in web development and engineering I think you could easily find a lab that would have a project where you could work on data analysis or database development, but it would likely be unpaid. I would email around, profs are often excited about engineers who want to do this type of work for them. If it goes well you could even get a letter.
I think your background in EE is great and they definitely do look for people with your background, but usually you would need some kind of demonstration of research interest in biology as well. I applied to UC Davis for computer science and UCSC for bioinformatics. I think they're both decently competitive, so if you have your heart set on going next year I would probably apply to a broader range of schools. Not that you wouldn't be a good candidate, depending on your statement, rec letters, etc, but it's always really hard to judge how the process is going to go until you try. Keep in mind that there is a fair deal of randomness to it as well.
For rec letters, if you had any professors/classes that you really liked you may be able to get better letters than you think. I would email your old professors and remind them of your performance in the class and send them an updated CV, hopefully they will agree to write one (they usually do). Having a publication is really great as well, they will look for that.
If you live anywhere near the bay area, it might be worth talking to some graduate students at Davis or Santa Cruz in labs you are interested in working in. You may learn more inside stuff about the programs this way and possibly also get a chance to meet some profs if you drop by the lab. I think if they know you before the application process and you're on their minds it increases your chances of getting in.