How to convert arcpy.InsertCursor to arcpy.da.InsertCursor in Python code?

The key difference between "classic" cursors like arcpy.InsertCursor and the new cursors like arcpy.da.InsertCursor is that the list of field names to search, update, or insert is now a required parameter, where it was only optional before. This parameter should be passed as a list of string objects. In order to search, update, or insert geometry, you need to use one of the SHAPE tokens: SHAPE@ will allow for full read/write access to the geometry of the feature.

For arcpy.da.InsertCursor specifically, there is no longer a newRow() method. Instead, you just pass a list of objects corresponding to the field names you specified in the arcpy.da.InsertCursor.

Here's how you would change your code:

array = arcpy.Array()
point = arcpy.Point()
tie_shp_file = arcpy.env.workspace + "/" + out_name
insert_cursor = arcpy.da.InsertCursor(tie_shp_file,["SHAPE@"]) #Editing only geometry field

for line in all_lines:   #all_lines is of the form array([[0, 3],[0, 7],[1, 3]) indicating line [from, to]
    #from
    point.X=coords_pts[str(pt[line[0]])][0]
    point.Y=coords_pts[str(pt[line[0]])][1]
    array.add(point)
    #to
    point.X=coords_pts[str(pt[line[1]])][0]
    point.Y=coords_pts[str(pt[line[1]])][1]
    array.add(point)
    polyline=arcpy.Polyline(array)# Create a Polyline object based on the array of points
    array.removeAll() #Clear the array
    insert_cursor.insertRow([polyline]) #Pass list with polyline to insert cursor. 

You can see how the [polyline] list used in the insertRow method corresponds to the ["SHAPE@"] list in the cursor's constructor. If you needed to insert additional fields, they would have to be passed in corresponding order.