This was the message on my phone today – "the cops are here, they think I have illegal pornography on my computer and want to search. Can they search it without a warrant? Should I talk to them?" And then the phone went dead. I suppose the caller had a little talk with the nice officers and they straightened it all out – or maybe not. Remember those TV cops who ALWAYS tell the obviously guilty suspect that he or she has "the right to remain silent" and how the suspects always "lawyer-up?" In real life what happens is the suspect almost ALWAYS talks, because we believe we can talk our way out of trouble.
That nice officer cannot solve the problem you face based on your explanation. So do not talk. Let me repeat – DO NOT TALK TO THE NICE OFFICER. Call a lawyer and have a real discussion – a confidential discussion – about your situation.
Back to the computer question – no, they generally cannot search without the warrant. The constitution bars unreasonable searches and a search without a warrant is generally held to be "unreasonable." That is not always the case as there are many exceptions to that rule – hence the reason to call the lawyer.
Can they get a warrant? Depends on what facts they have to convince a judge that there is evidence of a crime on the computer. Make them get a warrant and if they do, then they get the computer. But guess what happened in this situation? The suspect turned over the computer to the cops. They get the computer and whatever is on the hard drive. Will there be evidence of a crime? If so, there is no problem with the search because the suspect consented to the search and seizure.