Replaced GFCI no power in line wire but power in load wire

Hard to say exactly what the problem is. Pictures would help. But here are a couple of thoughts:

Overloads can cause multiple problems

It is possible that an overload at one receptacle can damage that receptacle and also damage wires in other locations. In particular, if you have a string of receptacles, something like:

• Breaker
• Receptacle - Living Room
• Receptacle - Bedroom
• Receptacle w/GFCI - Lanai

then an overload at the GFCI could cause problems in the living room or bedroom receptacles. This is especially likely if those receptacles are functioning as part of a chain - power in on left and out on right (or vice versa) and the connections are done with backstabs rather than screw terminals. If you had a weak connection before, an overload may have made it weaker.

Breaker Mismatch

"15amp breaker" plus "Leviton 20a dual pole" is against the rules and a Really Bad Idea. The other way around is OK - 15A duplex receptacle on a 20A circuit.

The problem is that when there is a 15A breaker, it is there to protect wire (normally 14 AWG) that is rated for no more than 15A. By installing a 20A receptacle, you are encouraging use of larger appliances (as was the case here). While the breaker should provide enough protection, using a smaller breaker than the receptacle can lead to nuisance trips and a temptation to swap the breaker (without making sure all the wire is 12 AWG or larger), as well as other problems. This should be a 15A dual receptacle, not a 20A dual receptacle.

If the load has power and the line doesn't, the GFCI outlet is connected backwards.
You figured that out, congrats.

The way it was connected, it didn't give you any GFCI protection. When you use the test button, the outlet should go dead.

You also mention

"the screws are a bit rusty from being outside outlets"

Are those outlets exposed to weather? If so they may need to be in more weather proof boxes, and possibly using UF wire, not NM. The NEC makes a lot of distinctions about 'damp' and 'wet' environments.

To settle some discussion about 20A outlets on 15A circuits:

"210.21 Outlet Devices

Outlet devices must be rated at least that of the load. (B) Receptacles. A single receptacle on an individual branch circuit can have a rating only equal to or larger than the rating of the individual branch circuit it is on except when installed in accordance with Section 430.81(C). ...(2). The ratings of receptacles where there is more than one on a circuit are shown in Table 210.21(B)(3)."