Since the name of the method is same, it is getting confused and displaying in the same placeholder.
I will recommend below structure, see if you would like to implement.
Create 2 place holders on one page:
Create one View Rendering with Textbox & Button and place it in SearchBox placeholder.
Create one Controller Rendering for SearchResult with one normal method & one Http Post method. Pass empty list of values for normal method
On click of the button in View Rendering, write a Ajax to call the method which is decorated with Http Post and which will pass the results to the controller rendering.
And in this case your Search Box will also be maintained.
Hope this helps and let me know if you have any queries.