are you sure 't' will change in the same proportion every season? If so, i think it would be ok to set t = 1,2,3,4,5...., or you may spend some time on 't' first.
i am interested in how you will do the seasonal adjustment. I think one way to test whether t is a stable trend could go for testing whether y(t)-y(t-1) is convergent.
I think your previous model is more straightforward.Seasonal Dummy Varable could be applied by assigning value of 1,2,3,4 for each period.