Yes, you're ultimately liable for damages you cause.
I would expect your insurance to ask the damaged party to release you from further liability as a condition of accepting settlement at the coverage limits, but if the damages are significantly over the coverage limits and it seems likely that they could significantly collect on a judgement beyond your coverage, it's a possibility.
Umbrella insurance can go up to much higher limits ($2M is maybe already be an umbrella policy), sometimes up to $10M is easy to get, and depending on your insurer, often the upper millions are much less expensive. The tail risk is pretty small. If you're worried about it, may as well ask for quotes at $5M and $10M.
I would expect your insurance to ask the damaged party to release you from further liability as a condition of accepting settlement at the coverage limits, but if the damages are significantly over the coverage limits and it seems likely that they could significantly collect on a judgement beyond your coverage, it's a possibility.
Umbrella insurance can go up to much higher limits ($2M is maybe already be an umbrella policy), sometimes up to $10M is easy to get, and depending on your insurer, often the upper millions are much less expensive. The tail risk is pretty small. If you're worried about it, may as well ask for quotes at $5M and $10M.