Is it OK to drop the periods in abbreviations?

Solution 1:

It depends on the type of abbreviations. USA is an initialism, and as such, does not require periods. The style of acronyms and initialisms no longer requiring periods is now becoming more prevalent.

Abbreviations, generally, still require periods (e.g. Prof. — Oh, and e.g.) Although there are exceptions, especially in British english usage, as noted in your linked related question.

Which abbreviations take periods and which do not is variable depending on the style guide followed.

Solution 2:

Lowercase initialisms like e.g., i.e., n.a., and q.v. generally retain their periods to help distinguish them from words (or, in the case of n.a. and relatives, inexplicably take a slash in the middle instead).