I don't know about its past usage, but here in the United States, at least,
'awkward' is not the same as 'odd', 'strange', 'peculiar' or 'bizarre'.
It's core meaning is the same as 'ungainly' , which translates in German as 'plump', 'ungeschickt' , 'unbeholfen' or 'schwer zu handhaben' be it either in action or simply in appearance. Something that's 'awkward' may indeed also be 'strange' or 'peculiar' or 'odd', but not necessarily.
An 'awkward sentence' or an 'awkward sentence structure' doesn't have to be 'peculiar', just inelegant in sound, ability to pronounce, ability to easily understand, etc.
A person described as 'awkward' conjures up an image of someone who lacks social grace, gracefulness in movement, be it either because of exceptionally long limbs, causing an appearance of lack of fluidity of movement or because of lack of social finesse.
For example, it's not uncommon for people to say: "I always feel awkward in social gatherings like parties."
This is much different from someone saying: "I always feel peculiar [or: odd or: weird] in social gatherings."
The second sentence expresses the idea that someone becomes overcome by inexplicable feelings or thoughts when in a social situation.
However, the word 'weird' has become quite a popular word on the street often meaning not much more than 'uncomfortable.'
An 'awkward moment' is a period of time during a social interaction which is 'embarrassing' (peinlich), but not necessarily 'unusual' or 'odd'.
Once again, the word 'weird' has taken on a catch-all meaning, so is often used in informal speech, particularly by young people with unfortunately limited vocabularies, to mean anything from 'different' to 'bizarre.'
So, informally you'll hear: "There was only one weird moment, but basically everything went smoothly." In this case the speaker means 'awkwardly embarrassing', not 'bizarre'.