I made some changes to this since it wasn't working correctly.  Would anyone be able to take a look at the changes I made?  I was able to change the code to insert and to update when it already existed.  I'm having issues now with inserting the contact record on already existing accounts.

Basically in the bottom part, in the first half of the update, I am looking to see if a Contact with Contact_Type__c = General exists.  If so, then don't create and add the id the contacts to update portion.  if a Contact with Contact_Type__c = General doesn't exist, then insert.  I thought i could use the isEmpty(), but then it says the variable c doesn't exist on the red line below.

trigger InsertUpdateGeneralContact on Account (after insert, after update)
{
    if(Trigger.isInsert)
    {
        List<Contact> ct = new List <Contact>();
        for (Account acc: Trigger.New)
        {
            //  Insert a contact on account insert with mapping

                 ct.add (new Contact(
                         FirstName = 'Account',
                         LastName = 'Email',
                         Email = acc.Company_Email__c,
                         AccountId = acc.id,
                         Fax=acc.Fax,
                         MailingStreet=acc.BillingStreet,
                         MailingCity=acc.BillingCity,
                         MailingState=acc.BillingState,
                         MailingPostalCode=acc.BillingPostalCode,
                         MailingCountry=acc.BillingCountry,
                         Phone=acc.Phone,
                         Contact_Source__c = 'Contact'
                         )
                     );   
            
        }
        if(!ct.isEmpty())
            insert ct; 
    }
    else{
    
        List<Contact> lstContacttoInsert = new List<Contact>();
        Set<Id> setAccountIdwithEmail = new Set<Id>();
        Set<Id> setAccountToInsert = new Set<Id>();
        Map<Id, Account> contactmap = new Map<Id, Account>{};
        
        for(Account objAccount: Trigger.new){
        
              setAccountToInsert.add(objAccount.id);

              contactmap.put(objAccount.id, objAccount);
             
             for (Contact cnt : [select AccountId, Contact_Type__c
                           from Contact
                           where AccountId in :setAccountToInsert
                           and Contact_Type__c = 'General']){
                           
             Account acc = contactmap.get(cnt.AccountId);
             
             //if(cnt.isEmpty())
             
             Contact c = new Contact(LastName = acc.name,
                        AccountId=acc.id,
                        Fax=acc.Fax,
                        MailingStreet=acc.BillingStreet,
                        MailingCity=acc.BillingCity,
                        MailingState=acc.BillingState,
                        MailingPostalCode=acc.BillingPostalCode,
                        MailingCountry=acc.BillingCountry,
                        Phone=acc.Phone);

                        lstContacttoInsert.add(c);
            }
        
            /*if(Trigger.oldMap.get(objAccount.Id).Company_Email__c == null && objAccount.Company_Email__c != null){
            
                //  Old value of company email was null and it was assigned a value, so create a contact
                
                lstContacttoInsert.add (new Contact(
                         FirstName = 'Account',
                         LastName = 'Email',
                         Email = objAccount.Company_Email__c,
                         AccountId = objAccount.id)
                     );
            }*/
            
            if(Trigger.oldMap.get(objAccount.Id).Phone != Trigger.newMap.get(objAccount.Id).Phone){
            
                //  Old value is updated, so collect the account ids whos contact is tp be updated.
                
                setAccountIdwithEmail.add(objAccount.Id);
            }
        }
        
        //  query contacts based on the set of account whose company email field was updated.
        List<Contact> lstContact = new List<Contact>([Select FirstName, LastName, Email, AccountId,
                                                            OwnerId, Fax, MailingStreet, MailingCity, MailingState, MailingPostalCode, Phone, Contact_Type__c
                                                            from contact where AccountId in: setAccountIdwithEmail
                                                            AND Contact_Source__c = 'Contact']);
        for(Contact objContact: lstContact)
        {
            //  Modify lstContact with Accounts company email field 
            objContact.Email = Trigger.newMap.get(objContact.AccountId).Company_Email__c;
            objContact.Fax = Trigger.newMap.get(objContact.AccountId).Fax;
            objContact.MailingStreet = Trigger.newMap.get(objContact.AccountId).BillingStreet;
            objContact.MailingCity = Trigger.newMap.get(objContact.AccountId).BillingCity;
            objContact.MailingState = Trigger.newMap.get(objContact.AccountId).BillingState;
            objContact.MailingPostalCode = Trigger.newMap.get(objContact.AccountId).BillingPostalCode;
            objContact.MailingCountry = Trigger.newMap.get(objContact.AccountId).BillingCountry;
            objContact.Phone = Trigger.newMap.get(objContact.AccountId).Phone;

        }
        if(!lstContact.isEmpty())
            update lstContact;
        if(!lstContacttoInsert.isEmpty())
            insert lstContacttoInsert;
    }
    
}