Possession of child pornography is a federal crime (and likely a state crime as well). Downloading the images into your computer is "possession".
Whether the govern would wants to prosecute you or win a conviction against you, no one here knows since no one knows your relevant facts. Win or lose, being prosecuted for child pornography is likely not a pleasant way to spend your time.
My understanding is that computers keep records of what other computers access files on the computers holding the files. Moreover, machines are identifiable unless the user takes steps to prevent identification. This is how the government is able to track down users of child pornography sites.
Moreover, data on computers are not actually erased unless spaces holding the data are overwritten by other data. Data that most people think are erased can be recovered by trained technicians.
There are "disk wiping" software that helps to overwrite the data the user wants to erase. Depending on the amount of data being overwritten and how many times the operation is performed, it may take several hours. My understanding is that for hard drives holding very sensitive data the government physically destroys the hard drives to prevent any chance of recovery.
The criminal defense division of my firm has dealt with this issue on many occasions in both State and Federal court. The authorities are way too busy with real child pornographers - people who promote trade and swap these images to go after someone who surfed onto it and then moved on. They focus on those who upload images and videos onto the web or who are choosing to download the images onto their computer. So I think you will be fine as long as you have avoided that conduct.
You should review your facts and options with an attorney. If you do not have enough income, you may qualify for free representation from the public defender. If your income is not enough but too high for free representation, the public defender association may offer you low fee representation.
Except with the advice of your attorney, you likely should not discuss the details of your case with anyone, especially the police. While the police is helpful in most situations, in this case, the police is not there to help you but to gain evidence to convict you.