再次查阅了一下朗文和牛津词典，两本词典对find一词的用法解释基本相似，观察所列举的用法条目，似乎可以找到一些思路。

1.《朗文高级英语词典》对find一词的释义及例句：

to discover that someone or something is in a particular condition or doing a particular thing when you see or touch them.

1) find somebody/something+adjective

I’m sure we’ll find her hard at work when we get home.

He tried the door and found it unlocked.

2) find somebody+prepositional phrase

She woke to find a man by her bed.

3) find somebody/something doing something

Often he found her quietly weeping alone.

4) find (that)

She looked at her glass and was amazed to find it was empty.

2.《牛津高阶英语词典》(纸质第9版) find词条7（为简练计，只列举用法条目，不列举例句）:

1) find sb/sth/yourself+adv./prep.

2) find sb/sth/yourself+adj.

3) find sb/sth/yourself doing sth

4) find (that)…

通过对朗文和牛津词典的例句观察，find sb/sth+adj./adverb/prepositional phrase/doing应该可以和find (that)进行同义转换，用that从句体现出一般体意义和进行体意义，或者静态意义和动态意义。以朗文词典的例句为例，应该可以这样：

1) I’m sure we’ll find (that) she is hard at work when we get home.

2) He tried the door and found (that) it was unlocked.

3) She woke to find (that) a man was by her bed.

4) Often he found (that) she was quietly weeping alone.

5) She looked at her glass and was amazed to find it empty.

如果按照这个思路解释:

C项：we found the lamp burning but the door shut.

可以转换为：

We found (that) the lamp was burning but the door was shut.

A项：we found the lamp being on but the door shut.

会转换为：

We found (that) the lamp was being on but the door was shut.(恐怕不能成立）

所以，倾向于选择C项，不选A项。

个人理解，未必正确，仅供参考。