Hi Ben,
For starters, the 30 watt iron maybe partially to blame -- the 40 watt weller is a great move.
Also, couple of things -- make sure the parts to be soldered are clean, and also clean the tip of the iron frequently.
The tip of the iron should be somewhat shiny, not dull, and you can use either a damp sponge, or if it's really oxidized badly, you can try plunging the tip into some bronze wool -- the bronze is softenough that it won't scratch and erode the tip. When the tip is nice and shiny, addd a bit of solder to it to tin it, and also to have a tiny bit of excess that will make contact with the parts to be soldered and heat them much faster -- the tiny blob of solder transfers more of the heat to the joint more uniformly and more quickly. Then touch the solder to the joint, and it should flow right on. Count to three once it starts to flow and then remove the iron tip and then don't let anything move until the solder fully hardens. It's quick, but will take longer with larger parts that hold more heat., and silver is another story altogether -- but you're not soldering solid silver.
Also make sure you have a flux or rosin core solder. If you do and this is still happening, you might be able to help the 30 watt iron out a bit by using some paste flux, but only apply this to the parts to be soldered, not the iron itself.
Give these things a try and if things are still not working, just wait until the 40 watt iron arrives. Also, the type of point on the soldering iron can make a difference, but try these things first.
Hope this helps,
Jim