If it resembles their characters itty is not "completely custom" rather it is more likely a derivative work in violation of 17 USC 106. many of these "characters" are trademarks, often for tee shirts, since those are popular souvenirs. Create your own character or you take a risk. How much of a risk? See an IP attorney for specific review and advice.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Whether you need a license or not would depend on the degree of similarity of your characters with their characters. You would certainly need to contact an intellectual property attorney specializing in copyright and trademark law to determine if your characters are okay to use on t-shirts and other merchandise.
I agree with the others. You need to get a proper and specific analysis. At the end of the day, if your customers are buying your products because they appreciate the branded characters depicted on them you have your answer.
I suggest that you consult with a lawyer in private and discuss your objectives in more detail. You can start by calling around to several for a free phone consultation, get some insights then pick the best fit to work with.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed with the law firm of Natoli-Lapin, LLC on the basis of this posting.