I've written the below code to update the field on case object when we're getting email to case. But when I tried to write logic to fire case assignment rule based on whatever value we're having in emailMessae.toAddress, it's not working at all.

Public class emailMessageHelper {
    Public static void updateCase (list<EmailMessage> newEmailMessageList, map<Id, EmailMessage> oldMap){
        Set <Id> caseIdSet = new Set <Id>();
        list <Case> caseList = new List <Case>();
        Map<Id, Case> caseMap = new Map<Id, Case>();
        list<Case> newCaseList = new list<Case>();
        
        for (EmailMessage messageObj: newEmailMessageList)
        {
            caseIdSet.add(messageObj.ParentId);
        }
        
        if(caseIdSet.size()>0)
        {
            caseList=[Select Id, OwnerId, Owner.Email, Status, status_Alert__c from case where Id in :caseIdSet];
            for(case caseObj: caseList)
            {
                caseMap.put(caseObj.Id,caseObj);
            }
        }
        for (EmailMessage messageObj: newEmailMessageList){  

            Case caseObj = caseMap.get(messageObj.ParentId);
            caseObj.Source_Email__c = MessageObj.ToAddress;
            System.debug('messageObj.ToAddress abc '+messageObj.ToAddress);
            if(caseObj.status !='New' && caseObj.Status_Alert__c != 'New Email Received' && messageObj.incoming && messageObj.toAddress!= null ){
                caseobj.status_Alert__c = 'New Email received';                
               
                Database.DMLOptions dmlOpts = new Database.DMLOptions();
                dmlOpts.assignmentRuleHeader.useDefaultRule= true;
                caseobj.setOptions(dmlOpts); 
                
                newCaseList.add(caseObj);                
            }          
            else if(!messageObj.incoming){
                caseobj.status_Alert__c = '';
                newCaseList.add(caseObj);
            }
        }
        if(newCaseList.size() > 0){
            update newCaseList;
        }        
    }
}

I'm calling the method in After insert and After update context.
Can any one please help me out on this?