Duration of selection and survival of the fittest
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The sentence until it evolves into a new species and / or the old ones become extinct sounds weird and there seems to be some confusion between evolution and speciation. Also it seems that you have a very static view of what is a species, as if species were predetermined categories into which evolution came to show itself!
I am not sure but I think your question has to do with "valley crossing", that is "how can a population evolve toward some phenotypic trait if intermediate phenotypic traits are not fit?" Is that your question? You may be interested in "shifting balance theory" but it might be a bit too advanced. 
It is common to overconsider the valley to cross (the deleterious impact of interemediate traits) and those traits are not necessarily so deleterious
Mutations do not always have small effects on traits. One mutation may have some very important impact and therefore, one mutation alone might be enough to cross a whole valley! You amy be interested in Fisher's geometric model. Fisher's geometric model is just some very arbitrary model and is not much inspired by any biological observation but it helps getting the concept of how mutations impact fitness (=function of survival and reproductive fitness).
When you say how do animals who are not well adapted get enough time to evolve to survive? I may use the word animals to refer to populations or to individuals in a population (extinction of genotype is a population and extinction of populations are two different things). If you referred to individuals then, your interest seems to be into "valley crossing" and "mutational effect"(effect of a mutation on phenotype or fitness). If you referred to populations, then you may be interested in "evolutionary rescue" or more broadly speaking in the so-called "theory of extinction".
There are probably other posts (or part of answers) on this site that will help you. A very good answer would be very very long. But maybe a moderately good answer may be relatively short!