15A Cartridge Fuses Keep Blowing When I Change Them

You might want to consider replacing the panel with circuit breakers. It sounds like the cartridge fuse circuits are overloaded or have an intermittent short

Of course you shut off stuff before replacing the fuses. Everything kicks on at once, what do you expect but a blown fuse. Sounds like you are overloaded, add some more circuits. To save money on fuses till you get the job done right, but those screw in circuit breakers. They screw right in the fuse socket, cost about $10 or less.

And, in reinforcement of what another poster said, just to make certain--

Turn off the _loads_ on the circuits, not just the power to the circuit panel. The problem is that apparently the circuits are so near capacity that the surge is blowing them.

Also as noted elsewhere, it does sound as though you are really in need of upgrading these circuits or adding additional or balancing loads from these to others that aren't so heavily loaded...one would expect to go years between blowing a fuse on a normally loaded circuit so if this is happening relatively frequently, it's a message.