The Hork-Bajir Chronicles, Chapter 2 header:
Andalite date: year 8563.5
Yeerk date: Generation 686, early-cycle
Hork-Bajir date: late-cool
Earth date: 1968
The Hork-Bajir Chronicles, Chapter 7 (Page 39 in my book):
<Yes. Dak... I am going to become a chadoo.>
He had no answer to that. I wasn't surprised. The morphing technology is so new that there are even Andalites who doubt its safety or usefulness. Fortunately, I had a friend back on the home world whose mother was one of the designers of the Escafil Device. She had shown it to me. I'd used it.
#34, The Prophecy, Chapter 6 (Page 36/37 in my book):
"Aldrea, don't say anything for a moment. Just listen. Let me explain," I said softly. When I felt her acceptance, I rushed on. "You were brought to this planet because there is a colony of free Hork-Bajir here. Your grandson, Jara Hamee, is part of the colony. So is your great-granddaughter, Toby Hamee."
(Some other relational ties are also known, to give an idea of lifespans: Andalite War-Prince Alloran-Semitur-Corrass was alive and active in The Hork-Bajir Chronicles, was disgraced due to the events of said book, and later trained Elfangor and Arbron. The Yeerk that would come to be known as Visser Three (by possessing Alloran's body) was also born at the start of The Hork-Bajir Chronicles.)
#18, The Decision, Chapter 24 (Page 136 in my book):
<We have few Andalites here on the planet now. And none who possess the wide array of morphs you have,> Galuit explained. <All Andalite warriors are morph-capable. But few acquire morphs or use them. That is mostly done by our people in intelligence. Spies. But you four may be able to penetrate the Yeerk forces.>
It took 5 years (1996 - 2001) to publish the series, and I vaguely remember a statement from one of the last few books that says that's roughly the same amount of time the Animorphs had been fighting the Yeerks. So since The Decision was published in 1998, that's presumably the year it takes place.
The above makes the Escafil Device around 30 years old during the series - likely to be one single Andalite generation, given the known relationships above.
Lastly, according to Galuit, Andalites don't morph very often - and those that do shouldn't be caught anyway. So non-Andalites are unlikely to have seen the morphing technology, making it "new" to most of them. And the way Galuit phrases his statements, I'm under the impression that the morphing ability also isn't well-known in the civilian population.