" 時々" means "sometimes" and "しょっちゅう" means "frequently", right? And, the meaning is well short of "いつも".
Now, I can't imagine "しょっちゅう" is used in writing or formal speaking, right?
So, what is the proper way to express the idea of "しょっちゅう"?
Yes, you are right that "しょっちゅう" is not appropriate in a formal context. As others said, "度々（たびたび）", "頻繁に（ひんぱんに）", "よく" are good in formal contexts. I would like to add "繰り返し（くりかえし）" and "何度も（なんども）" as more options when you want to express that something happens repetitively.
Let me give an example in which "しょっちゅう" sounds wierd. Suppose that a head of a company comes up to a press conference and apologizes about a sequence of accidents revealed by a journalist. If he says, "社内で調査しましたところ、同種の事故がしょっちゅう発生していたことが判明いたしました。申し訳ございません。", that sounds strange. He should say, "社内で調査しましたところ、同種の事故が度々、発生していたことが判明いたしました。申し訳ございません。", or he could replace "度々" with "頻繁に", "繰り返し", or "何度も". In this example, "よく" does not fit well.
As another example, if you want to say "I am sorry for calling you many times.", "しょっちゅうお電話差し上げて申し訳ございません。" sounds strange. The mode of formality is obviously different between "しょっちゅう" and the rest of the sentence. You should say, "度々お電話差し上げて申し訳ございません。". You can replace "度々" with "何度も" or "繰り返し", but "頻繁に" sounds a little strange somehow. In this example, "よく" does not make sense.
As yet another example, if you want to say, "I often came to this library when I was a student." in a speech upon the closing ceremony of the library, then you will say, "私は学生のとき、この図書館によく来たものです。" but not "私は学生のとき、この図書館にしょっちゅう来ていました。" unless you want to present yourself as a relaxed and friendly person.
As a final example, if a policeman writes an official report stating that a suspect has visited the victim's house frequently, he will write, "被疑者は被害者の家を頻繁に訪れていた。". He can use "何度も" and "繰り返し" in this context. It is strange to see "しょっちゅう" in such an official document as well as in an academic thesis, a news paper, etc.
Japanese has many levels of formality and writing is not necessarily formal by definition. Let's look at a thesaurus:
屡屡 ・ 再再 ・ 往往 ・ しげしげ ・ 屡々 ・ 始終 ・ 頻繁 ・ 間間 ・ 再三 ・ 度度 ・ 間々 ・ 良く ・ 頻く頻く ・ まま ・ 頻って ・ 往々 ・ 比々 ・ 許多度 ・ 繁繁 ・ 度々 ・ 重重 ・ 頻りと ・ 繁々 ・ たびたび ・ 頻りに ・ 時時 ・ 比比 ・ 重ね重ね ・ 繁く ・ よく ・ ちょこちょこ ・ ちょいちょい ・ ちょくちょく ・ 数多度 ・ しきりと ・ しきりに ・ しばしば ・ 屡
Hmm, let's see たびたび (度々)is definitely formal so maybe that's a good word--but people don't really use it in the same way they use しょっちゅう. It's used in a lot of set or almost set phrases though.
頻繁に is quite commonly used:
About 12,400,000 results (0.37 seconds)
as is its noun 頻度. And could be nice additions to your options.
On the other hand I don't think there is anything necessarily wrong with しょっちゅう or よく（良く） especially in formal conversation, in writing then sure you might prefer 良く or 頻繁に over しょっちゅう, but there are lots of types of writing and I'm not even convinced there.