Sounds like a weak battery to me. It will power the lesser current draws (interior lights, door locks, ect) but the heavy draws (blower motor, starter, headlights) will not work well.
Replace the battery.
'05 3.0L/Auto, AWD, 226k, Hella 550 lights with 100W bulbs mounted to the front bumper, $15 mudflaps on both ends, homemade 2" receiver hitch, A/C wont hold charge anymore
'05 Dodge 2500, Cummins/48RE, QCLB, 4wd, some aftermarket goodies
'12 Buick Enclave, Premium AWD
'07 Honda VTX-1800T, tons of add-ons including electronic cruise control
'96 Honda XR400R, 440 kit, head work, cam, carb, pipe, home brew street legal kit
'70 Chev C-10, currently being restored, to be powered by TBI 454/4L80E