If by dragon goby, you mean any form of Mandarin or other dragonet, no they won't help with clean up. All they will do is eat all the copepods that came on your live rock and then die unless you have a refugium. Dragonets pretty much only eat live near-microscopic crustaceans. They would be simple to keep, but a normal tank simply cannot produce enough of these creatures for them to eat, so they wipe out the whole population and then starve to death. A rare few have been successfully trained to brine shrimp or other simple to produce food, but most need copepods, and it would cost you at least $20 per month to keep buying live copepods and having them shipped to you.
For starters... You have two damsels, so you may not be able to get much in the way of peaceful fish, because damsels are little war machines. They attack any fish that enters their territories. Some of the bigger ones attack sharks. When you add more fish, you'll either want to rearrange the rocks, pull out the damsels for a few days, trap the damsels under something like a colander for a while, or stick to only somewhat aggressive fish that can handle damsels like dwarf angels. The idea is not to add peaceful fish that will get beaten up because they don't know the territory while the damsels do. Ideal would probably be to pull out the damsels, rearrange the rocks a bit, add the other fish, wait a couple of days and add the damsels. Generally, you have the least trouble if you add your fish so that the most aggressive ones go in last, because fish are much less aggressive to fish that were in a place before they were.
Next rule: only add fish at a rate of 1-2 per month. Why? Because adding them too fast causes your tank to cycle again, which will kill any fish less tough/mean than damsels and torture the damsels, making them meaner and more prone to illness.
Lastly, follow Cearbhaill's advice on quarantining. Healthy fish + sick fish = more sick fish. Sick fish become dead fish and then you end up needing to leave your tank empty anyway, so leaving it empty longer at the beginning is a small price to pay.