I won't mention the ground wires, but hook them up also or just leave them connected, there won't be any reason to futz with them. If disconnecting a ground wire improves something, that reveals a ground fault problem, which should be fixed.
- Note how everything is connected, not just wires, but how the wires are grouped in cables. Generally you have a Romex cable coming in with a black and white, and sometimes also a red. The red is an alternate "hot".
You will always use hot and neutral as married pairs, except for switch loops. Think of the hots and neutrals as monogamous married couples that socialize in pairs, i.e. Jim's wife only hangs out with John's wife if Jim hangs with John. The grounds are like their kids who all play together.
Disconnect all the wires from the GFCI.
Hook up only 2 wires to the GFCI, the hot and neutral from only one source.
Test the GFCI thoroughly. If it does not work perfectly, repeat 2-3 until it does. The goal is to find which of these sets is a reliable power source.
For each of the remaining pairs of wires:
If you don't want that pair of wires to be GFCI protected, pigtail them with the wires now going to the GFCI's LINE inputs. Test the GFCI and the other outlets again. There's no reason this should make the GFCI fail.
Any pairs of wires you DO want protected by the GFCI, attach them to the LOAD output terminals of the GFCI, one set at a time, and run the GFCI through its paces. If attaching any wire pairs causes a GFCI trip, that branch has a problem, and you will not be GFCI protecting it until you solve that problem. (it could be a faulty appliance, that is, after all, what GFCIs detect.)
Once all the wires work, connect them all to the GFCI LOAD terminals, using pigtails unless the GFCI has room to take all the wires on its LOAD terminals.
The key to complex connections to GFCIs is "one step at a time".