I just had exactly the same issue. After much forum searching, I replaced the ignition switch module - problem solved. Part was $43 at NAPA. Installation was very easy - there are a few threads and even a couple of YouTube videos that show how to change it out.
'98 TJ Sahara - 6 cyl. 5 spd. | Rancho 3" Lift | 32x10.5 pro comp | Centerline wheels | Turbo City, Tomken, Skid Row armor | Warn i8000 | Airaid intake | DIY rear swingaway and bumper
'02 Chev Z71 Avalanche | '06 BMW R1200GS | '02 BMW F650GS Dakar | '11 Husaberg FE390