Yes, they will draw twice the current.
How do you plan to use these lights? Will they be always-on, or part of an animated/sequenced song where they will be at constantly changing dim levels? If the latter, you will likely not have any issue.
Even if always on, I think the wires are heavy enough (smaller AWG or gauge, means thicker wires) to handle the increased current. Remember, for LED lights, the string normally carries 20ms to 30ma. Even doubled this is less than 50ma and the wires should be able to handle a couple hundred milliamps. To be absolutely sure, you must determine the gauge of the wires, then look up the wire capacity.
I cut one of my strings (actually, it was already cut). The (stranded) copper wire has a 0.026" diameter. That means it is about 22 AWG. At 60 °C that should carry 3 AMPS. That is well above the doubled current of your fullwave conversion. I would not think that wires to be a concern.
You might want to check the resistor "size". Not necessarily the resistance, but the heat dissipation. Remember that dissipated heat is W (watts) = I^2*R (Current (Amps) squared times resistance (ohms)) Doubling the current makes the heat go up by a factor of 4. If it's a concern, you could replace the resistor by same resistance, higher wattage rating. Again, constant on could be an issue, but varying dim levels should be a concern.
Jimboha
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