Replace them with screws.
The best way to fix your situation is option 4: remove the nails and use hangers with or without screws.
Nails, (also known as Spike and Ferrule) are an old school way to hang gutters, which is hard to see nowadays, usually you'll see it in older homes, (if the gutter is still hanging) and just simply in my opinion are a bad way to hang gutters that will not last long.
Gutters will expand and contract with heat / cold. This will make for the nail system to weaken and eventually fail.
Hemmed gutters are supposed to be "free-floating" to allow for expansion and contraction, but for time saving purposes, pretty much everyone uses gutter hangers and screws to the back of the gutter. Straight back gutters are designed to be used with hemmed hangers. The advantage of screws vs nails, is that screws are much stronger because of the threads, and are much better at not allowing the gutter to move as much depending on the weather.
You should really consider using gutter hangers with screws or if the screws are not long and / or thick enough (since you're likely going in an existing hole), get hangers without the screws and use your own screws.
Hangers have hooks that hook onto the face of the gutter (hem) and are also press formed / stamped, wider, which makes them much stronger, and usually pretty cheap, you can even get them at Amazon.
I would definitely stay away from nails.
Nail vs gutter hanger with screw and without screw.