2002 Dodge Stratus only starts when it wants to

Only 180,000 miles and giving me trouble! every few days i turn switch to start car and it blows a fuse (sometimes radio, sometimes turn signal, sometimes cigar lighter)and starter does not engage. no clicks, no nothing. replace blown fuse and cranks right up. Dodge cannot explain why starter disengaged when fuse blown. Dodge checked starter, voltage regulator, ignition switch, battery, fuse boxes, alternator, cannot find anything wrong. says bring it back when not working, so i did…towed it in last time it would not start (that time it did not blow any fuse). Dodge took it off tow truck and it started right away! at wits end…help!