The problem is twofold:
1) The crystals are just a simple MCA, so the TEC is only stabilizing the diode. Most of the mode/power instability in a DPSS laser happens in the crystals so in that regard these and the vast majority of simple, inexpensive lasers will always exhibit some instability.
2) The driver could be lousy quality. I don't have an analog version of this module, but I do have several TTL models. If the driver isn't linear (through crappy engineering), then that would explain the dotted lines and other anomalies as well.
To help determine which is the cause, try feeding the laser TTL signals and see if you still get all of the same symptoms. If things seem generally better when running TTL signals, the driver is the most likely culprit.
Mine do fairly well with TTL, so I think that they should be pretty acceptable in terms of crystal stability, but of course each will be different.