It's not likely that closing your company would end the matter, unless the plaintiff isn't persistent and simply loses interest. It sounds from your question that the plaintiff is suing you only in the name of your company. However, it's not clear from your question what form your company is in -- it sounds like it's likely a sole proprietorship (as contrasted with, for example, a corporation or LLC). Assuming it's a sole proprietorship, closing your company likely would have little effect on the lawsuit, as it could be amended or (as you suggest) re-filed to name you individually; the dba company name is really identical with you, and your assets stand behind it, as a practical and legal matter. But what if your company is a corporation or LLC wholly or mostly owned by you? In many respects a corporation or LLC does afford protection against personal liability for business debts. But legally, at least in the state where I practice, if you close a company and transfer any corporate or LLC assets (e.g., computers, furniture, inventory, cash, etc.) to an individual without paying the company for them, you're subject to a legal action based on "fraudulent conveyance" under the NY Debtor and Creditor Law. A lot depends on the specifics of your situation, but as a general rule, you stand a good chance of being at least partially on the hook, potentially. Please note that I practice in NY. As they say, "Your Mileage May Vary." Best of luck.
If your company is a sole propriertorship and not an LLC or a corporation, then you are generally held personally liable for its debts and liabilities. In most states, there will be no need for the plaintiff to dismiss and refile the case; he will simply need to make a motion before the court to amend his lawsuit to name the new defendant or you, personally.
If your company is an LLC or a corporation, you cannot simply close up shop and reopen next door under a different name. Most courts in this situation hold the new company to be simply an "alter ego" or a mere continuation of the prior company and would readily allow an amendment to name the new company. Worse yet, you may expose your company or yourself, individually, to liability for attempting to defaud creditors or unduly delaying the litigation.
If your company has been sued, you should consult an attorney to determine whether to mount a defense or to attempt to settle the lawsuit on favorable terms.
DISCLAIMER: I am not your lawyer. No attorney - client relationship is formed here. This answer must not and should not be relied upon as legal advice, rather, it is offered for general informational purposes and as a starting point for further consideration.