Sounds like you have 2 issues here.

1. Neutral Safety Switch
2. old dirty oily starter

My W123 300D did this when I got it, the starter would continue cranking until it killed the battery. The starter had 23 years of gunk and oil and dirt inside it, causing the armature to stick, either not engaging or engaging and sticking. Cold weather made it worse.

I replaced it and the sticking issues stopped.

__________________
91 350SD
14 F150 Eco
19 Fusion Hybrid
11 GT500