As a cash basis taxpayer (which is most common for self-employed businesses), you report income in the year received. If you received the income in 2018, you should report it in 2018.
You should tell the person who issued you the form 1099-misc that you will only be reporting what you were paid during the year (as required) and they need to correct the form. This is the easiest thing to do. They should correct the form. Doing it early makes it easier for them to fix too.
Otherwise, keep records of when you were paid and report the income in 2018. The IRS or state may reach out to you and you should reply with a statement, explaining you weren't paid until 2018, that you requested the form to be changed and your proof that you weren't paid until 2018.
For proof, be sure to save screenshots of and downloads for your bank statement and invoice, payment processor or whatever software they paid you through.
