If my company is being sued and I’m incorporated, am I able to shut down that company and set up another with a slightly altered version of the same name?

Generally speaking, if a corpration is being sued, the owners can dissolve it and escape liability. (There are certain exceptions: e.g. for personally guaranteed debts, certain tax debts, tortious acts if the owner him/herself committed the tort--e.g was the driver of a company car that hit someone).

However, what you describe may pose problems. If you shut down one company then promptly open another to carry on the same actions, under a very similar name, a court could conclude that you are attempting to perpuate a fraud on creditors; or "pierce the corporate veil" to impose liabilty directly on you in some cases; or impute the debt of the first company to the second, on the ground that they are essentially, for all equitable and practical purposes, one and the same.

The courts do not like it when someone clearly tries to manipulate the system to deny others their rights--therefore, while if you simply shut the company down and went about your way, you'd likely escape liability, what you suggest could easily be seen as a fraudulent act. When there is a fraudulent act, the courts have what's called "equitable" powers to fashion a remedy to alleviate the unfairness; therefore, if the plaintiff is aware of the new entity and what you did and seeks relief from the courts, there is a reasonable chance the new company--or you personally--could be held liable.