Hi, there -
I was just poking around 'My Forum' and came across your question, Jan ... And I'm wondering, can anyone chime in? I hope if I am overstepping, it is helpful overstepping!
Essayer is a "first group" verb (it ends in -er), you're absolutely right, and it follows the usual conjugation rules for that group. However, I like to think of the stem of essayer as irregular, because it has variations such as the one you found.
As I recall from my university studies years ago, though I just started Rocket French lessons, essayer follows the rules of other verbs that end in "-ayer" (example payer). Because it ends in "-yer", essayer may be conjugated either by turning the stem "y" into an "i" as you did, or by leaving the "y" as it is spelled in the lesson. In both instances, one must then conjugate onto the verb stem with the same ending that usually applies to those first verbs that end in -er, as seen in your example/question.
To try to sum it up, if a verb ends in "-yer", keeping the "y" and conjugating the verb or changing the "y" to an "i", then conjugating the verb are both correct/accepted. You'll see this variation in the future, conditional, and subjunctive (well, sometimes, ugh) tenses of these "-ayer" verbs, of which there are probably, I dunno, 30 or 40. They're common. Yay, right?! Doesn't that make these verbs SO much more fun? At any rate, I listed the full future (simple) tenses for essayer below in case my written explanation was hard to follow:
j'essaierai OR j'essayerai
tu essaieras OR essayeras
il essaiera OR essayera
nous essaierons OR essayerons
vous essaierez OR essayerez
ils essaieront OR essayeront
... If you care to see "essayer" in all of its conjugated glory, Lawless French https://www.lawlessfrench.com/verb-conjugations/essayer/ has it done on 3/4 of a page. I'm not sure how Rocket will feel about my forwarding the link, but it is just a really easy 'at-a-glance' to get a handle on some of the conjugation nuances that may not be detailed in our lessons (suggested supplement, Rocket, not a promotion, I swear).
Hope this helps and I haven't stepped on any toes! I'm headed back now to the relative bliss of my Level 1 refresher lessons.
Best wishes for success!