Hi Ryno,
Given that it was your responsibility to inform the department of your change in mailing address, and you failed to do this, I don't believe you have grounds to dispute the fines issued and not received. The fact that the toll fine has increased to $250 is due to interest on unpaid fines. Further, is there a way online to check when your eToll credit is running low? If you knew, or ought to have known by checking online or paying your eToll pass regularly (e.g. monthly pass), then moving address would again, not be a reasonable justification.
Did you only give the department your mailing address? Did they further have your email and/or telephone number?
Have you attempted to contact the department and explain your situation/excuse and see if you can have the extra fees waived in exchange for you paying the due toll fines?