Under no circumstances should you accept the 2,000 miles offered. You were involuntarily denied boarding on a flight which actually operated, and on which you had a confirmed reservation. This is covered by EU regulation 261/2004 which details your rights in this situation. It is irrelevant what the reason they did it was. They breached the rules by not seeking volunteers in the first instance. If it was done "in error", you are due compensation anyway.
You are entitled to €600, for a flight not within EU of greater than 3500 km in distance as compensation for your inconvenience. This is without prejudice to any other claim you may make.
I would not wait for the outcome of the "investigation". I would call Continental and tell them that you are pursing your rights under the EU regulation and that you believe you are entitled to compensation of €600 under EU law. Unless they make payment to you within 30 days you will file a claim in Small Claims Court. You can file your claim electronically and will not have to appear in court. The cost of filing your claim is very small, in your case a maximum of 10% of your claim, up to £120. You will get this back when you are successful.
You can file your claim at
www.moneyclaim.gov.uk Write to Continental to confirm you are claiming your rights under EU 261/2004 immediately after talking to them on the phone and confirming your demand for compensation within 30 days. Send this letter by recorded delivery, which will cost you about £1. This requires them to sign for the letter and will act as proof in court that you exercised your rights under the regulation.
Good luck.