ECM, VAR, dummy and dummy interaction term

[lenasong7]

Dear all,
Can someone please help me? I'm a beginner and not sure if I'm doing correctly since this is my first time learning VAR... especially with dummies!

I'm trying to estimate the following regression: Price = C + B1*Cost + B2*Dummy + B3*(Dummy*Cost) + u, where Dummy is equal to 1 for some periods from say years 200X~20XY, and 0 for periods before 200X. This is to check any change in price and cost relationship after the end of collusion.

I tried Unit Root test, and found that the time series data for Price and Cost are nonstationary,
but after doing the normal OLS regression* (with ar(1)<- due to serial correlation... ) and trying unit root test on the saved residuals, I found that u ~ I(0). So there's cointegration here.
Now I'm stuck at this point, uncertain what to do next.

I understand that when dealing with nonstationary series, one must worry about spurious regression.
So that means, I cannot just take OLS regression of the equation above and report the coeffs right?

So I tried to estimate VAR. I understand that dummy is put as exogenous. But what about Dummy interaction term, i.e. Dummy*Cost?
Isn't this interaction term endogenous? But then, I would get Dummy(-1)*Cost(-1). Is this correct?

And some coefficients of the obtained short run ECM is not significant. Is this possible and ok?
When rearranging the long short run ECM and substituting lag operator L = 1, I would get a long-run model, which coefficients are different from those initially obtained above (*). i.e. the OLSE when I just did with nonstationary variables in the beginning.
When reporting the final estimation values, which one should I take as the long run equilibrium equation?

Please please give me any advice.
Thank you!!!