It's not that your LLC would be held liable under the contract: you would be. If the contract stated that you shall not participate in or own or benefit, etc. from a business delivering similar services, you cannnot do that under the guise of an LLC, either. The LLC would not be sued--it was not a party to the contract--but you would be; and the business suing you could seek both monetary compensation for your breach and a court order (injunction) barring you from working for the competitor. The contract you signed obligates you and restricts what you may do, regardless of the business structure you attempt to work under.