From the mouth of an SPD adult.

I'm nearly twenty, a mother of a 2 year old and I was diagnosed with SPD later in life (by later, I mean I was around 16 years old)

The signs were all there, however.
No seams on socks, no vacuum, big crowds made me nervous, no mixing my food, EVER, because more than one texture was horrible. No mud, no clay, no chalk.

Nevermind the fact that I had a blanket with silky edging that I cried for weeks when my father removed said edging. Weeks. I was probably four years old.
And I chewed on EVERYTHING.

Some got better as I got older. I was able to mix my food. Sometimes. I can wear regular socks..as long as I fix the seams a bit. I can use a vacuum.. I just prefer not to.
Silly things like peeling potatoes has me washing my hands for 10 minutes. But I still eat potatoes..often!

I definitely hate big crowds, but I will go somewhere with them if it's something I want to bring my daughter to. Though I may avoid big lines. I really don't like when people bump into me.

And I now have a shirt I've dissected with scissors to make perfect sized bits of silky fabric to play with during high-stress times.
I never had any therapy. By the time I was diagnosed, the damage had been done. I failed multiple art classes and a woodworking class because I couldn't be near the clay in art or the machinery when it was running in the woodworking.
However, the point being is I did make it through school. I did make it as an adult with SPD, without any therapy. Without anything, really.
And, thankfully, my daughter seems to be SPD-free.
So keep up hope :) It is likely your child will be A-OK in the end.