Check the connections to the neutral and reverse switches, left side down by the hi/low/rev shifter. Take all the connections apart, clean them, and reconnect. Check all your grounds as well. The usual culprit is a bad connection or short. The neutral light is turned on by the neutral switch grounding the wire to it completing the circuit, so if it was on all the time it could have been a bad switch but that doesn't fit since both are now dimly lit. I would go through all the connections.
Another thought you will have to consider... It might be time for a new CDI unit. The CDI could be causing the dimly lit indicators and only starting with the pull start as well, i had that happen.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....