Else = Otherwise
The question of grammaticality is yes. OED sense 4 covers this, and it has four subsenses:
a. In another case, under other circumstances; otherwise, on any other supposition; if not.
b. preceded by or. Also OR ELSE, with aposiopesis (the alternative to be imagined), as a colloq. form of warning or threat.
c. idiomatically. = ‘If it is not believed’. Now rare or dial.
d. qualifying an adj. rhetorical.
And here are a couple of its citations for 4a:
- 1837 J. H. Newman Par. Serm. (ed. 3) I. v. 115 ― Else how should any one be saved?
- 1873 Browning Red Cott. Night-C. 115 ― Boughs above, Darken, deform the path, else sun would streak.
So else can indeed start an independent clause, in which case it means otherwise. This is grammatical.
Else = Elsewise
In this sense, else is equivalent to elsewise, which per the OED means:
In some other manner; in other circumstances, otherwise.
One of the citations for using elsewise in this way is from the last novel of Charles Dickens:
1865 Dickens Our Mutual Friend I. 97 ― Elsewise the world got up at eight.
So you could use elsewise instead of just plain else, but that one might get you talked about, since some people don’t realize it’s even a word. Then again, apparently you already get flack for using else in a grammatical way as though it were some sort of violation of grammar when it is not.