I had similar problems when I started. A couple of times I had new queens that were so bad I lost the hives. Sounds like the bees want to replace yours. I've had more success with supercedure than introducing new queens into a poor hive. You'll find minimal resources coming into the hive not only because conditions are bad outside, but because the hive can't afford to send many foragers out. Sometimes I've found introduced queens are less likely to take in a weak hive. Perhaps the queen is not laying well because not much is coming in. Feeding syrup might help a little bit, but I haven't had much success with doing that either except to keep them from starving.
Ideally, it would be good to have a frame or two of brood to kick start the population. If you know another beek, they might be able to help you out. Are bees covering most of the 10 frames? if they are only on 3 or 4 of the frames then it would be better to reduce the size of the hive, either by moving frames to a 5 frame nuc box, or putting in a board. They cope better with reduced space and also defend against pests better.
Have you been around to see what flowers are out? It's dry here too, but some trees flower despite the dry. You could probably stop feeding when your population increases, as it would be rare in the SE not to have enough foraging plants unless you are surrounded by cropping land. Syrup doesn't help pollen stores in any case.
In short, since they have supercedure cells I would not introduce a new queen, though that's up to you. Michael Bush's bee maths should help you work out how long to expect a new laying queen. It's advisable not to disturb them while they are making queens. http://www.bushfarms.com/beesmath.htm
However...there's a chance the old queen will destroy the queen cells. I would probably just wait and see what happens. Sometimes it does take a queen some time to start laying properly..it just can be hard to break that cycle of low population, low stores and low brood. Might be it's a learning experience. I'm sure you'll find a big swarm or cut out and have a thriving hive before too long.