The unit is up and running again. Here is what I did to get it working:
I called Haier to comfirm what PCB to replace. I first spoke with someone in parts who was no help at all. I called back and got tech support and spoke with a fellow who was great. He told me that the PCB is able to detect poor air circulation through the condenser and that this will cause a code 13 to show. He also said that this was a common problem on these units due to a build up of lint in the condenser (makes sense since there is not a lint trap.)
I had noticed a few darks spots in the condenser but didn't think that it would be enough to cause a problem. I decided to pull off the fan/heater housing so that I could access the top of the condenser, and I also pulled the drain hose from the bottom of the condenser. When I pulled the hose, there was a large plug of lint right at this junction. Also, I was able to pull quite a bit more out by using a piece of wire and running it through from top to bottom and vice versa. Once I had done this, I got my garden hose with the spray attachement hooked up to it and blasted it down the top of the condenser to try to give it a "final rinse". There was quite a bit of lint that came out, including a second large "plug". At some point I remembered that the fellow from another post (who was having the same issue as I was, and had the same unit) had said that in addition to changing the PCB, he pulled out a lot of lint from the condenser...
Since the condenser had such a build up, I decided to open the fan/heater housing to make sure it was in good condition (since linty air flows through there also). There was some lint stuck on all fins of the fan, but overall, it was in pretty good shape. I doubt that this would have caused any issues. I cleaned it up very well anyways, and put it all back together. Et voila!
I learned a few other things from the tech support guy worth mentioning: the PCB knows when the water has drained because it is able to sense how much amperage the drain pump is drawing while in use. Once the water level is low, and the pump is pushing some air, the amperage decreases to a certain level and the PCB knows that the water is drained. The PCB is also able to sense the "balance" of the load. If the laundry is all balled up, it will continue trying to make it more balanced before allow it to get up to a high speed spin, which needs to take place before moving on with the cycle. The drain pump is running during all of this, and is normal. It is common for balance issues with only one piece of clothing in the unit (which is what I was doing when experiencing what I thought was a draining issue (but really was a balance issue, I think).
I have now run a couple of loads with no problems.