The reason there are so many verbs being used is that they're all doing a lot of work together to describe something hard to describe.
Part of the nature of my professional work is to listen very carefully to how people describe experiences they've been through, particularly how they describe their inner subjective experiences of things they have done or others have done to them. I've noticed that some people lean very hard on verbs to convey their inner experience. These people tend not to have a lot of practice or experience using elaborate noun vocabularies for emotions or other subjective experiences. So they "make do" with the verb-heavy idiom of expression they are most familiar with.
In the example you bring us:
All I can do is sit and worry about [the] grilling I'm going to get from....
The thing that is being communicated is the felt sense of futility or helplessness. "All I can do" is an idiom that means "I can't do anything except", and "sit and worry" is an idiom that means "worry", but in a particular way (more on which below).
So right from the start, we have three verbs all used in conjunction to communicate the idea of not doing anything – and feeling bad about it!
And then, in the second part, we have grilling, going, and get. Now, "going to" is synonymous here with "will" and is not functioning as a verb in its own right. Now, the speaker could have said they were worried about "being grilled", but the choice to phrase it as "a grilling" they're going to "get", they idiomatically suggest more aggression on the part of the griller. To be grilled, by default just means to be asked a bunch of questions in a way which is demanding that one account for oneself. To get a grilling suggests the grilling will be punitive, the way one "gets" a punishment, or a bad attitude, or a dressing-down.
All of this adds up to the speaker saying that their worry is that this other party will do something to them they don't like (grill them).
So what this whole utterance boils down to is, "I'm anxious [the unspecified other party] will feel angry with me." Two verbs: "am" (be) and "feel". One for the speaker and one for the other party.
But because the speaker, for whatever reasons, does not express themselves with adjectives like "anxious" and "angry" (or nouns like "anxiety" and "anger"), they are left expressing the idea in verbs, making heavy use of idioms that employ verbs to convey feelings.
Which brings us to "sit". This idiomatic use of "sit and [verb]" (e.g. "sit and think", "sit and stew", "sit and stare") uses sit figuratively to mean "not take external action". It establishes a contrast between outwardly observable behavior (little to none at all) and inner behavior (in this cases, intense mental activity).
So "sit and worry" means, "do nothing outwardly while churning emotionally inwardly". In that sense, it's somewhat redundant of "all I can do is".
We have someone saying "all I can do is", which means "I can't do anything except", and "sit and worry", which means "do nothing outwardly and ruminate inwardly on my worry"; and then we have them saying that the thing they're worrying about, the "grilling I'm going to get", which means "the hostile confrontation with which (I anticipate) this other person will greet me".
While there are shorter, more concise ways of expressing the same thing, the use of all these verbs in idiomatic ways conveys pretty clearly to the astute listener – presumed familiar with the idioms in question – the inner experience which is actually the subject of that sentence, and it does so with quite a bit of nuance. It is not merely that the speaker feels worried, but that they feel despairing: they anticipate they are about to be confronted hostilely, and there isn't anything they can do to stop it.