Can I personally get sued by someone who fell at my home in addition to them suing under my homeowner’s insurance?

They do not  sue "under your homeowner's insurance"--they sue you, if they believe you were at fault in causing their injury (see below). If you have insurance, your insurer should defend (e.g.  pay for a lawyer), indemnify (pay any amounts you are determined to owe), or even settle the case for some lesser amount the plaintiff, or person suing you, will accept--at least up to the limits of your insurance policy. (An insurer does not need to spend or pay one dime more than your policy is for.)

Therefore, what happens is that the person injured can make a claim for a certain amount of money, which you will presumably pass onto your insurer. If they claim for more than your insurance coverage, or your insurer will not pay (such as because they believe you are not liable; or because  they feel this accident is not covered for some reason), the injured person can sue you; they can sue you for any amounts not paid by your insurer.

If they win--which means, if they can prove that you were at fault in some way, such as by creating or allowing a dangerous condition, like, for example, rickety stairs, to exist (you are only liable if you are at fault; you are not liable solely because someone was injured at your home)--then they can get a judgment against you for an amount equal to the costs and injuries they can prove you caused. You would, as discussed above, have to personally pay any amounts not paid by your insurer.