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I asked my teacher about this once, and the impression I got from her answer was that it's just a generic thing put in there to make it more polite. It can also replaced with さん.

Checking Denshi Jisho returns this definition -

2: used (gen. in fixed expressions) to make words more polite;

どうもありがとうございます。

Yes, it is true that today it only used in fixed expressions anymore, but there is more. A native speaker would not know its original meaning and it would only feel like "just a generic thing put in there to make it more polite." However, this does not explain how the さま originally got into these expression.

If you take a look at its kanji form 様 you might recognize this is the same kanji used for よう as in for example 関わらないようにする. And indeed, if you look it up again on jisho, you will find

3: manner; kind; appearance

which is indeed this kanji's original meaning*. Given this, expressions such as おつかれさま uses さま to express "It would seem your are tired." which is undoubtedly more polite (and more meaningful) than saying "You are tired." It is also a less direct manner of speaking.

Oh, and さん is actually a shortened and thus less honorific form of さま, which explains why one can also say おつかれさん. Speakers nowadays are not aware of the original of the さま part in such expression and so come to confuse it with the address suffix さま. for which they know さん is a more casual form.

* Note:
Actually, the very first meaning of 様 when this character was created appears to be that it

suggests the luxuriant foliage of a certain variety of tree, a horse-chestnut or oak, the nuts and bark of which were a staple food and dyeing product, respectively, in certain parts of China.

but that was a long, long time ago and perhaps even before it was imported to Japan and before it could be said to mean anything in Japan. Read it up here: http://www.kanjinetworks.com/eng/kanji-dictionary/online-kanji-etymology-dictionary.cfm?kanji_id=ONM34

As for さま being an address (suffix), I would have guessed it were a polite way of addressing by saying "the appearance of [person]", however, the mentioned site also explains that

Mr./Mrs./Miss/Ms. is a borrowed usage, deriving from the old Japanese term そおも → さま, meaning "over there."

The usage of さま meaning manner is used heavily in dictionary definitions.

どうもありがとうございます。

Can i can translate 「関わらないようにする」　as "Try not to concern youself with".