Have you cleaned and/or tightened the big, main battery cable connection at the starter solenoid? If I am interpreting the clues correctly, there is a faulty connection somewhere.
When the dash lights come on, will they stay on indefinitely, at least until you add more load, like trying to crank the engine over?
When the dash lights go out, when do they come back on? When you let the key return from “start” to just “on” or what?
If you dont perform these simple tests, folks can’t help you. If these tests are beyond your skill or experience, you might be in over your head and need to call in someone who could get some answers for us.
BTW, a fusible link is either good or open. There is no in between. Just like a glass fuse-it either carries full power or it doesn’t carry any at all.
A poor, loose or corroded connection, on the other hand, can and will carry SOME power, but maybe not enough to meet demand.