Just to be clear, are you saying that half of the total sample size is in the accountability group and the other half is in the non-accountability group (and within each of those, they are split into one of the three categories for competition?)?

Would you clarify what you mean by a scale variable (I think I'm unfamiliar with that particular term).

Assuming that the dependent variable is appropriate for an ANOVA/Regression, I agree with your supervisor that it would be better to fit a regression with the appropriate dummy variables.

First, you will need to determine the baseline/reference group. I recommend setting low competition and non-accountability as your baselines. In other words, for competition, you have three groups, so you will need 2 dummy variables (number of groups minus 1, and fit the intercept in the model). Create X1: 1 if medium competition, 0 if not; Create X2: 1 if high competition, 0 if not; this implicitly codes low as the reference level (0,0), so if your hypothesis #1 is true, you would see positive coefficients for X1 and X2 since they represent the difference of the group relative to low.

Create X3 for accountability: 1 if accountability, 0 if non-accountable

To test for interaction, include terms: X1*X3, X2*X3

You will have 5 coefficients aside from an intercept.

Conduct a global f-test and proceed if significant (also make sure assumptions are reasonably satisfied).

I would then do a nested f-test to test the 2 coefficients for interaction.

If you've followed up until this point, feel free to post some output with the full model and interaction subset test. If you're a little confused, feel free to ask some more questions!