The answer to your question is state specific. In Connecticut the answer to your question would be no. You need both thye corporate/LLC shield as well as the e&o insurance. They protect you against different liabilities. The e & o insurance will protect you  against negligence claims.

In Connecticut corporate officers/directors and managers/memembers of LLC's are personally liable on their torts regardless of whether they were perpetrated in their personal or representative capacity. In most cases e&o insurance will not cover intentional torts.

As an alternative to suing you for negligence your adversary may choose to sue you for contract breach because it has a longer statute of limitations. In most cases e& o insurance will not cover contract breach claims, but the corporate/LLC shield will limit your liability to corporate assets and exclude your personal assets from satisfaction of the claim. However, you will need to be careful when you sign the contract to completely and correctly identify the name of the corporation/LLC as well as to identitfy your job title and clearly indicate that you are signing the contract in your representative capacity.