Per comments, OP's instructors are probably just incompetent (or perhaps he somehow misunderstood what they meant). In standard English, ***and*** has two distinct meanings...

> 1: *Dick liked Jane **and** she liked him*  
 2: *Dick insulted Jane **and** she slapped him*

...where #1 simply uses ***and*** to join two "independent" statements - stripped of pronoun substitutions,...

> 1a: *Dick liked Jane*  
 ***and*** (another statement, with no specific relationship to the first)  
 1b: *Jane liked Dick*

But in #2 the conjunction implies a temporal/causal relationship between the two statements...

> 2a: *Dick insulted Jane*  
 ***and*** (*subsequently* - pragmatically, probably *consequently*)  
 2b: *Jane slapped Dick*

Grammatically/semantically, the ***and*** in #2 can be replaced by ***and then*** or simply ***then***. Perhaps OP's instructors felt he was overusing the two-word form in contexts where it was pragmatically obvious that the second event was *later in time and/or caused by the preceding event*. But this is a matter of style, nothing to do with grammaticality.


TL;DR: Even if OP's instructors knew what they were talking about, they've obviously failed to convey the correct information to their student. The only reasons to avoid ***and then*** are (1) - if the context implies no [con]sequential relationship, and (2) - because using it *excessively* can result in a stilted prose style.


EDIT: In fairness to OP's instructors, they might be (mistakenly, imho) attempting to impose notions derived from *prescriptive grammar*. Consider...

> 2a: *Dick insulted Jane*  
 2c: *Then Jane slapped Dick*  
 2d: *Jane then slapped Dick*  
 2e: *Jane slapped Dick then*

...where 2c-2e are all valid sentences/clauses that could follow 2a. It's irrelevant here whether we put a period or a semicolon between 2a and whichever of 2c-2e we choose to use (you wouldn't hear a difference in speech anyway). The point is the fact that ***then*** can be moved around like this implies it's an *adverb*, not a [*coordinating conjunction*][1]. As that link says...

> *Be careful of the words **then** and **now**; neither is a "coordinating conjunction", so what we say about coordinating conjunctions' roles in a sentence and punctuation does not apply to those two words.*

I cant say I find the above caveat particularly enlightening, but it may help explain some of the antipathy directed at the usage ***and then***.