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The lifetime of a star is billions of years. There might be a species that has had to flee the death of its star, and there is a slight possibility of a species that has had to flee the death of a second star, but a species that has had to do this over and over again, and in the process filling up the galaxy? The universe isn't old enough for that.
Fermi's paradox doesn't involve billions of years. One of the claims that Fermi's paradox makes is "If interstellar travel is possible, even the "slow" kind nearly within the reach of Earth technology, then it would only take from 5 million to 50 million years to colonize the galaxy." That would involve species sending out new colonies, and those new colonies in turn sending out more new colonies in a very short period of time. It is more likely that intelligent species that survive for long periods of time don't have an overwhelming compulsion to expand, and instead live within their means. Our solar system contains a vast amount of resources to exploit.
I don't know that alien species would be any more inclined to beam an intense signal at us just in case we might be listening. We have sent out signals, but not for very long periods of time, and then only sporadically.
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