Solution 1:

Certain verbs are what we call subject-control verbs. This means that when they take an infinitival or gerund-participle clause as a Complement, we understand the Subject of the verb in the non-finite clause as being the same as the Subject of the verb in the matrix clause (the main verb).

Let's look at an example:

  • Bob wants the elephant to eat the donut.
  • Bob wants ____ to eat the donut.

In the first example, the noun phrase the elephant, which directly precedes the string to eat, should clearly be understood as the Subject of the infinitival clause. However, in the second example there is no noun phrase preceding this to-infinitive. If you like, the noun phrase that we might expect here has been deleted. It is represented in the example above by a space. Now we understand that the Subject of the verb eat here should be understood to be the same as the Subject of the verb want. We understand the second example like this:

  • Bob wants [himself] to eat the donut.

Generative linguists often talk about the phenomenon above in terms of Equi-deletion.

The example above uses a to-infinitive, but we also see this with verbs that take gerund-participle clauses as a Complement too:

  • Bob minds the baboons being late.
  • Bob minds ____ being late.
  • Bob minds [himself] being late.

The Original Poster's example

  1. *I felt being dragged off by a beast. (ungrammatical)
  2. *Bob felt being dragged off by a beast. (ungrammatical)

The Original Poster wonders why sentence (1) above is not grammatical. The Original Poster is obviously very familiar with subject-control type verbs in English. Now if FEEL was a subject-control verb, we would expect the sentences above to be interpreted like this:

  • I(i) felt [ __(i) ] being dragged off by a beast.
  • Bob(i) felt [ __(i) ] being dragged off by a beast.

If this was the case we would understand the semantic subjects of the infinitival clauses as me and Bob respectively.

However, sentences (1, 2) cannot have this meaning, because FEEL is not a subject-control verb. Sentences (1) and (2) are both ungrammatical. Writers like Bas Aarts, in Modern English Grammar (2011) describe the structure of sentences with verbs of perception, such as HEAR or FEEL, like this:

  • I heard Bob being tickled by the elephant.
  • I heard Bob(i) [ ___(i) being tickled by the elephant ].

In the sentence above, the verb heard is taking a Direct Object, Bob. We interpret the Subject of the gerund-participle clause through the Object of the verb hear (not the Subject). It follows from this that if there is no Direct Object of heard, we will not be able to interpret the gerund-participle clause properly, and it will be ungrammatical. If we insert a Direct Object into sentences (1) and (2) above, we will be able to interpret them like this:

  • I felt myself(i) [ __(i) being dragged off by a beast].
  • Bob felt himself(i) [ __(i) being dragged off by a beast].

We could represent those sentences like this:

  • I felt myself [myself being dragged off by a beast].
  • Bob felt himself [himself being dragged off by a beast].

So if we want to use verbs like FEEL in this way, we must include an explicit Direct Object. If not, we won't be able to interpret the Subject of the following gerund-participle clause.

How about past participles?

Now, the verb FEEL in the sentences above is a verb of perception. That means that we have used our senses to directly perceive something happening. However, we can use FEEL with a different meaning. We can use it too talk about how we are physically or emotionally feeling. Here are some examples:

  • I felt happy.
  • Bob felt euphoric.
  • The elephants felt underappreciated.
  • The boxer felt battered and bruised.

In the sentences above happy describes I, not an event. Similarly euphoric describes Bob. Underappreciated is a description of the elephants' feelings. An in the last sentence battered and bruised describes the state of the boxer. Notice that these words do not describe some event that is being perceived. We cannot construe happy as a Direct Object of the verb felt in the first example. It is completely different from the sentences:

  • I felt the tremor.
  • I felt the earth tremoring.

Here the tremor and what is represented by the earth tremoring are external events that are perceived by me. We could think of them as events which are witnessed by the Subject. But in the examples further above, happy and so forth describe the state of the Subject. These words are not Direct Objects, but Predicative Complements. They just describe the state of the Subject. Notice also that these words are adjectives, not verbs. The last example uses the phrase battered and bruised. We can show that this is an adjective and not a verb because we can use the adverb very to modify it. We can't usually use very to describe verbs:

  • It is very big.
  • I felt very battered and bruised.
  • *I very like it. (ungrammatical).

Now we probably could use some kind of participle as a Predicative Complement here. However, it would need to describe a state, not an action. The predicative Complements that FEEL takes in sentences like this describe states and feelings, not actions We cannot use the participle dragged here. The sentence:

  • I felt dragged by an animal.

Is probably not grammatical. If it is, it means something like I felt like I had been dragged by an animal. It definitely cannot mean I felt that an animal was dragging me, or I felt it as an animal dragged me. It is easy to understand why a non-native speaker might be tempted to use a past participle with this verb. However, the examples that they are hearing which seem to use participle verbs are almost definitely mostly adjectives not verbs.

Solution 2:

It's bad writing to begin with, If one has to justify its meaning or explain to a reader what it should mean. In my opinion, these are better options that don't leave a reader scratching their head:

1) I felt myself being dragged by a beast.

2) I felt as if I were being dragged by a beast. (in this example you might want to say what you were being dragged through or over, as most people cannot relate to what being dragged by a beast is like. To me it would be preferable to be dragged over a grassy field than a forrest floor.)

In any case, this is a sentence taken out of context. In my opinion the context is necessary in order to make a decision about the correct grammar to use. To me, "I felt being dragged by a beast." seems like a fragment that needs more explaining or completion.