I am being sued for an accident that the plaintiff claims I was at fault.

The most urgent thing here is the clock, not the merits — in Texas, once you're formally served with a lawsuit, you generally have until 10:00 a.m. on the Monday following the expiration of 20 days from service to file a written Answer, or the plaintiff can take a default judgment against you without you ever getting to contest fault, damages, or anything else. That deadline runs regardless of what your insurance situation looks like, so don't let sorting out insurance delay getting an Answer on file.

Second, and just as important: report this to your insurance company today if there's any policy that could apply (your auto liability policy, most likely, but check homeowners/umbrella too depending on how the accident happened). If you have applicable coverage, your insurer owes you a duty to defend — meaning they're contractually obligated to hire and pay for an attorney to defend you in this lawsuit, typically at no out-of-pocket cost to you, even if the claim against you turns out to be weak or ultimately loses.

Where it gets more complicated, and where you may want your own personal attorney in addition to whatever lawyer your insurer assigns, is in a few specific situations: if the plaintiff's claimed damages could exceed your policy limits (called excess exposure) — in that scenario, the insurer-appointed attorney's client is technically the insurance company, not fully you, and if your insurer fails to settle within your limits when it reasonably could have, you could end up personally on the hook for a verdict above your coverage, so independent counsel watching that dynamic on your behalf matters; if your insurer sends a reservation of rights letter or denies coverage outright, meaning they're defending you (or not) while reserving the right to later say the policy doesn't cover this — that's a direct conflict of interest and a strong signal to get your own lawyer; or if it turns out you don't actually have applicable coverage for this claim at all, in which case no one is defending you but you, and you need your own attorney immediately.

So: get the lawsuit paperwork together (the citation showing when you were served is what starts the deadline clock), call your insurer today to put them on notice, and in parallel, have an attorney look at the suit and your policy to tell you whether the insurer-assigned defense is enough or whether you need your own counsel alongside it.