Wiring LED Lights Together and Running off of 3V Power Supply
Each letter most likely consists of a number of LEDs wired in parallel to the voltage source; itself two 1.5V cells in series for 3V total. Ideally, each LED will have its own current-limiting resistor, but sometimes they are omitted to reduce costs (such as with coin-cell powered devices).

AAA cells (like any alkaline cell) have internal resistance. Per the Energizer datasheet, this is 150-300 mΩ. In theory this means a maximum current, albeit for a very short time, of \$\frac{1.5V}{0.15Ω} = 10A\$, far more than an LED can withstand. This suggests to me that the product should have current-limiting resistors somewhere inside. You might want to inspect and post a photo to be sure.







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If this is the case, replacing the batteries with a line-powered 3V source should work. You could take some measurements with a multimeter to determine this more reliably.

To connect the letters together, you're correct that they should be in parallel:







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The switches you have are likely for 12V use, and probably already have a current-limiting resistor for the internal LED. If they are lighting at all with 3V you can simply connect the two LED terminals on the switch in parallel to the letter it controls:







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