I have a VF page including:

        <apex:form >
            <apex:pageBlock mode="edit" id="searchText">

                <apex:pageMessages />
                <apex:pageBlockSection >
                    <apex:pageBlockSectionItem >
                        <apex:panelGroup >
                            <apex:inputText id="searchTerm" value="{!searchTerm}"/>
                            <apex:commandButton value="Search" action="{!search}" rerender="results" status="status"/>

                        </apex:panelGroup>
                    </apex:pageBlockSectionItem>
                </apex:pageBlockSection>

            </apex:pageBlock>
         </apex:form>

        <apex:actionStatus id="status">
            <apex:facet name="start">
                <apex:outputText styleClass="searchword inputbox" value="searching... for '{!searchTerm}'"/>
            </apex:facet>
            <apex:facet name="stop">
            
            <apex:outputPanel id="results">

          ............displays the search results here................

        </apex:outputPanel>

        </apex:facet>
    </apex:actionStatus>
<br/>
debug -- {!searchTerm}<br/>

and a controller

public with sharing class SearchController {

    public SearchController() {
        searchResultLists = new List<Content_Item__c>();
        searchTerm = 'search...';
    }

    public List<Content_Item__c> searchResultLists { get; set; }

    public String searchTerm { get; set; }

    public void search() {
        for(List<Content_Item__c> results : search.query('find :searchTerm in all fields returning Content_Item__c'))
            searchResultLists = [select id, name, synopsis__c,
                URL_Link__c, Image__c, Release_Date__c, Heading__c, Content_Type__c,
                content_type__r.name,
                Content_Type__r.Id, Content_Type__r.Title__c, Content_Type__r.Show_Date__c,
                Content_Type__r.Image_Position__c, Content_Type__r.Image_Border__c,
                Content_Type__r.Read_More__c, Content_Type__r.Read_More_Dest__c,
                Content_Type__r.List_Length__c, Content_Type__r.Synopsis_Length__c,
                Content_Type__r.List_Style__c,
                Use_Attachment__c, Attachment__c, (Select Id From Attachments limit 1),
                (select id from Contents__r limit 1) from Content_Item__c where id in :results];
    }

}

I also have the following in a header component that renders through the template for the page:

       <apex:form >
        <div class="clearfix search">
             <div class="search-box">
                 <apex:inputText id="searchTerm" alt="search..." styleClass="searchword inputbox" value="{!searchTerm}" maxlength="20"/>
                 <apex:commandButton action="{!search}" styleClass="button" rerender="searchText" status="status"
                     image="{!URLFOR($Resource.fusionexperience, 'images/searchButton.gif')}" />
           </div>
       </div>
    </apex:form>

Both the page and the component reference the (same) custom controller in their header.

Now I'm getting two problems:

The first is that although entering the search text in the header component and clicking the command button triggers the controller action, the input text never seems to be passed to the controller and so the search fails.

The second is that although entering the search text in the main page and clicking its command button, does pass the search text and return the correct results, the debug line at the bottom of the page and the searching message in the facet don't return the expected  {!searchTerm} string - it's always the intialised value 'search...'

So what am I doing wrong?