If-statement: Best way to handle layering with same condition

I'm pretty new to programming, but I've been trying to make sure I learn good design practices. My question is related to how to handle this sort of if-statement situation, as it seems to violate Don't Repeat Yourself.

I have a class with a constructor that includes a method to connect to a database. The method will write to a string if there was an error in the connection code block. I then have a process method that analyzes metadata of the database, and will also write errors if any are found. I don't want the metadata analysis to run if there was already an error in the connection method, but is this the best way to do this?:

public bool Process()
{
    if (ErrorLog == null)
    {
        //Metadata analysis code that may write errors

        if (ErrorLog == null)
            return true;
        else
            PublishErrorLog();
        return false;
    } 
    else
        PublishErrorLog();
    return false;
}