This question comes up every now and then. The fact is, most cars from that era, including the Henry J and Allstate, do not have a fuse box. The 1951/52 Henry Js have the main fuse (30 amp) mounted on top of the headlight switch and most (but not all) electrical components run off a terminal on the headlight switch. The 1953 J, and probably the '53 Allstate as well, has an inline fuse on the main wire which connects to the hot terminal on the headlight switch. Radios and heaters were options on the J (and Allstate) and they have their own inline fuses and get their power from the hot terminal on the ignition switch (they use the hot terminal because the J, and many cars from the era, didn't have an accessory position on the switch.) If your car has the original wiring, what you're describing sounds like a grounding issue as many terminals and contact points have corroded in 60 years and if someone disturbs or moves them, the connections can be compromised. The headlights tie to the 3 terminal junction right behind the hood latch. Also, the headlight switch terminals can crack at the rivets and cause the lights to go dim, and in some cases, not work at all so your problem could also be the head light switch if wires were pulled and disturbed in the process.