Differentiating between verb-ing and gerunds [duplicate]

In sentences like, "I'm dying to get to you and "I'm learning to live again" and "i was preparing to go for surgery when you called" what are "dying", "learning" and "preparing" functioning as? Are they verb(ing) or are they gerunds.

It isn’t altogether clear what you're really asking here, because “gerunds” are always ‑ING verbs in the first place. Here’s how it really works. All verbs have a completely regular inflection ending in ‑ING. The thing is that such words can be used in two completely different ways: either they’re still verbs or they aren’t.

1. The first way you can use ‑ING words retains their properties as verbs, like being the head of a verb phrase, taking verbal arguments like objects or other complements, and being modifiable by adverbs.

   • earning a living
   • being happy
   • casually running a red light
   • always giving him a hard time
     
     

   These are still verbs when you do that even when you use the whole verb phrase as a substantive (something the Latinists call a “gerund”, but that’s not a great term in English) or as a modifier (something the Latinists call a “participle”, ditto).

2. The second way you can use ‑ING words strips them of their properties as verbs. When you do that, the result becomes another part of speech, usually a noun or an adjective. It is no longer a verb, so it can no longer take objects or head a verb phrase.

   If it’s a noun, it can be made plural, take adjectives, or be connected with prepositions. Other possibilities than nouns exist, like adjectives or prepositions.

   • lifetime earnings
   • during the day
   • the running of the bulls
   • owing to his earlier troubles
   • alien beings from outer space
   • my aching bones

In all your examples, those ‑ING words are still verbs, so yours are all of the first type not of the second. They are being used to create verb phrases with the progressive or continuous aspect.

The OP needs to look at both answers so far, by @KarlG and @tchrist, for the full picture about ways of using -ing verbs.

There are two ways of using -ing verbs that retain their properties as verbs, in progressive constructions and as gerunds.

Your examples all use progressive constructions.