In New York, there is no absolute requirement to appeal a negative outcome prior to bringing a legal malpractice action, nor is there a requirement "not to settle" a case prior to bringing a legal malpractice case.  The standard is whether plaintiff was "effectively compelled" to settle.

Here is a Florida Case, Technical Packaging v. Hanchett on the issue of whether an appeal is absolutely required prior to bringing the legal malpractice case.  They wanted to sue, and consulted with the target attorney.  Target attorney told them that the statute of limitations was 5 years, when in fact it was 4.  Target attorney turned down the case, and plaintiff hired other attorneys, only to find out it was late in starting the case.  Now, target attorney defends on the basis that plaintiff did not appeal the outcome. 

Read the case for Florida reasoning, but the outcome is that no appeal is absolutely necessary.