Is "too" not only an adverb, but an adjective as well?

The adverbs 'too', 'also' and 'as well' have the same meaning and are interchangeable. 'Too' and 'as well' normally come at the end-position, and 'also' with the verb in the mid-position in the clause.

A similar situation, as in the question here, has been dealt with in Michael Swan's Practical English Usage.

1- ( Other people have meeting on Sundays, and ) we have meeting on Sundays as well.

2- ( We do other things on Sundays, and ) we have meeting on Sundays as well.

3- ( We have meeting on other days, and ) we have meeting on Sundays as well.

These Adverbs of Focus ('too' and 'as well') can refer to different parts of a clause, depending on the meaning. When we speak, we show the exact meaning by stressing the word or expression that they refer to. In the first sentence, we say that we also have meeting like others; in the second sentence, we say that we have other things too to do ; and in the third sentence, we say that we have meeting on other days as we have meeting on Sundays.

The answer from @mahmudkoya might be more complicated than you need. Here's a simpler explanation.

In all three of your examples, "too" modifies the verb. It just takes a little getting used to to notice these common adverbs that modify verbs:

I stood up too.

I jumped first.

I arrived late.

I stirred it slowly.

In all these cases, the word in bold is an adverb that modifies the verb.