Once you get your new plastic handle and get the hood open. You can of course ensure that the battery cables are in good shape and tight. Also check the primary wire to the starter to ensure that it has a good crimp from the wire to the connector. You, of course, should always check connectors and wiring before purchasing parts.
However, the symptoms you describe indicate, in my experience, that you're solenoid and/or starter are the culprits.
Because the primary wire to the starter is relatively light and the current comes through the ignition switch, they are stressed, as the solenoid gets old and tired causing the solenoid current to increase. Many years ago, I took the primary starter wire and connected it to the primary side of a heavy starter relay, the kind found on most all Fords. I then connected a heavy cable from one of the secondary terminals on the relay to the positive battery terminal. I then connected another heavy cable from the other secondary relay connection to the primary side of the starter. This provides a heavier connection from the positive side of the battery to the solenoid. It saves the ignition switch and provides a more substantial connection to the solenoid. It's worked well for me for many years.
'84 Euro 240D, 516K miles
'88 300E 5 Speed
'81 300D Daughter's Car
Over 800,000 miles in