Verified answer

What's the difference between -ющий and -ая forms? The -ющий (-ущий, -ащий, ящий) ending words, indeed, act like adjectives that are derived from verbs (they impute a characteristic to a noun, the English analog is present participle). For ex., 'Отдыхающие на берегу туристы обязаны убирать за собой мусор'.

The -ая (-а, -я, -уя, -юя, -яя) ending words act like adverbs - they don't characterize the noun but the action. They are used when two parallel actions are happening to modify/specify the main action (the English analog is 'adverbial participle'). For instance, 'Собирая чемоданы, я нашла потерянное кольцо'.

Basically they are used for the same purposes, just their syntactic functions are different. You can say both 'Проезжающие мимо машины так и норовили меня облить' :) and 'Машины, проезжая мимо, так и норовили меня облить' - and it will basically mean the same thing. They are both translated with the -ing verb form in English.

Just on a side note: The adverbial participles (-ая, -уя, etc.) are *always* used with the noun / pronoun. You can't say, for ex., 'Переезжая с одного места на другое, у меня закончились деньги'. Only: "Переезжая с одного места на другое, *я* потратила все деньги'. So your sentence 'засыпая, держались руки сильно' contains a bad grammar mistake, because it doesn't have the noun which the word "засыпая" would be attached to ('hands' cannot fall asleep, right?:). It should be, for instance, 'Засыпая, супруги крепко держали друг друга за руки'.

Another note: these -ая and -ющий forms are features of literary language and hardly ever used in conversational language.

In this case, засыпать means 'to fall asleep', that is, to lie, drifting away, planning to sleep :)

Then, засыпая is 'while falling asleep'. 'They held their hands while falling asleep' is the whole sentence.

Летающий and спящий are qualities of the object - one object that is flying and another object that is sleeping; flying and sleeping respectively.

However, there's a difference between спать and засыпать. Засыпающая кошка is neither a sleepy cat nor a sleeping cat, it's a cat that is in the process of falling asleep, that intends to sleep and is almost sleeping.

Although, if she'll do something else at the same time - move her ears, for example - then you can use засыпая: Засыпая, кошка шевелила ушами. While falling asleep, the cat moved her ears.

You can even use both words in the same sentence, although it'll sound funny:

Засыпая (1), засыпающая (2) кошка шевелила ушами -

While falling asleep (1), the cat that was falling asleep (1) moved her ears.

Another example, with летать:

Летая над городом, можно увидеть его целиком - One can see the whole city *when (or while) flying* above it.

Летающий над городом человек может увидеть его целиком. - A person that *is flying* above the city can see it whole.

I hope it makes sense >.<