"He explains that since the at-fault party's car is so old that it may not carry enough coverage to cover even my medical bills."
With all due respect to the lawyer that told you this, this makes NO sense at all. Liability insurance covers damages caused by the insured to OTHERS. The quality or age of the car that a person drives might be a relevant factor in determining whether or not to carry collision coverage and what sort of deductable would be appropriate, but those things have no direct relation to the amount of liability coverage a person carries. As you probably know, CA law requires liability insurance of at least $15k per person/$30k per incident. While there is probably some correlation between a person driving a 20 year old car and an inability or lack of need to carry more than minimum coverage, there certainly is no direct relationship between the two.
"I have since sent a demand letter for policy limit disclosure to the other party with some of my medical findings. They have not responded."
The other party has no obligation to disclose the terms of his policy to you until and unless you sue him. That said, I certainly agree that it is a good idea to try and find out how much coverage he has (either directly from him or from his insurer).
"Is there any way to put my medical bills on a lien without an attorney?"
You can ask your providers about this.
"Is there any other recourse?"
I'm not sure what you mean? Other than what? You can either settle with the other driver and his insurer or you can sue.