I'd follow your lawyer friend's advice.
As somebody who's only watched a smattering of Judge Judy type shows, it sounds like a slam dunk case to me. Just remember to bring ALL of your paperwork in.
1. There should be an escrow contract, or at least an escrow portion where they set out all the legal obligations per your escrow account. This document says that they'll handle your real estate taxes. Ergo, they're supposed to pay them.
2. Bring the evidence that the taxes were not, in fact, paid per the above contract. Include documentation per the best of your ability of them NOT assuring you that they'd get the taxes paid by the end of the month. Is any of this in writing? Are you a two party state for recording? Get them on record.
3. Bring in the state notices as not only evidence that they didn't pay, but why it's causing extra harm to you.
4. I'd sue for 'extra expenses'. Keep it reasonable - day's wages for arguing in court, any penalties the state is charging, court fees, etc... Don't ask for the moon(don't piss the judge off), but if you can get an extra $2-300, so much the better.
5. Be advised that they might pay things very quickly once notified that they need to appear in court, and in many states they're forbidden from sending a lawyer to appear in small claims.
6. I'd still advise, like sumpnz, that you refinance to a less shady company. Though I have heard that in at least one case, after screwups like this the court voided the escrow portion of the loan since they proved that they couldn't manage to pay 2 simple bills(home insurance and taxes). They were forced to refund the entire escrow amount(no interest for them), pay the past due bills out of their own pocket, and from then on the home owner wrote the checks himself.