The only verb form of “obligation” that is traditionally considered correct is oblige, not “obligate”, so you cannot make a mistake by only using obliged and avoiding “obligated” altogether. That being said, there are a few things to consider.
The form “obligated” is almost never heard in formal British English, and it is usually considered to be an Americanism. However, in American English and colloquial British English, “obligated” can be quite commonly heard in place of “obliged”.
Some dictionaries of American English tag “obligated” as a synonym for “obliged”, but this does not seem to agree with actual usage; the word “obliged” has two different meanings, but only one of them can be expressed by “obligated”.
“Obliged” can mean “having a binding obligation”, “being required to do something”, as in
In this sense, “obliged” can be (and commonly is) replaced by “obligated” in American English:
However, “obliged” can also be used to express thanks to somebody who did something for you. It expresses your perceived moral obligation to do something for them in return:
“Obliged” is not normally replaced by “obligated” in this case, even in American English. If you speak or learn American English and are not sure which one to use in a given situation, go for “obliged”, which is acceptable in either case.