I've used the 7 watt LED equivalents in a few of my old beer signs, and never had a problem. I have seen the LEDs in the same package that were NOT designed for AC. Make sure the bulbs you got are for 110VAC, and then test them before install to make sure they work. By testing, you can check out your theory about hot and neutral being switched. You should try the LED bulbs you have now to see if it makes a difference.

If your plug is original, it did not matter which side was hot and which side was neutral, so flip the plug 180 degrees and try that. It was much later when they polarized the AC plugs so hot was always hot and neutral was always neutral. I doubt that's the issue, but you never know for sure until you try.