Am I using the subjunctive mood correctly?

No. With a counterfactual ("as though", in your example) you use "were", not "was".

From Wikipedia:

The English subjunctive also occurs in counterfactual dependent clauses, using a form of the verb that in the indicative would indicate a time of action prior to the one implied by the subjunctive. It is called the past subjunctive when referring counterfactually to the present, and is called the pluperfect subjunctive when referring counterfactually to the past. It occurs in that clauses following the main-clause verb "wish" ("I wish that she were here now"; I wish that she had been here yesterday") and in if clauses expressing a condition that does not or did not hold ("If she were here right now, ..."; "If she had been here yesterday, ...").

If you want to use the subjunctive, you should use were; if you do not, the mood is not subjunctive:

He spoke as though he were the only one to tell the truth.

Wikipedia's article on the subjunctive mood, in the section entitled To express a counterfactual hypothesis, states:

[T]he past subjunctive is used following the conjunctions as if and as though to express a contrary-to-fact situation that reality is supposed to resemble.

• She looked as though she were going to kill him.

As though can also precede was and still be perfectly correct. Again, grammatically speaking, this would not be in the subjunctive mood. See this forum thread:

There is only one situation that requires the irrealis (subjunctive) form: conditional inversion. Were he weary, he would not be walking so fast. Here "was" is not possible. In all other circumstances, "was" is an acceptable standard alternative to irrealis "were": as though he was already weary of the day is what I would normally say, though I could also use "were". It does not matter how hypothetical or unreal the situation is: that, if it does anything, merely increases the likelihood that the irrealis form is used. It does not make it obligatory.

You will use were when describing unreal situations, which is the case in your example sentence. As I interpret your sentence, he is not the only one to speak the truth, but he spoke as if he were.

The subjunctive form of to be is were. Thus, you would say, "If I were a doctor, I would not refuse patients." Here, I am not a doctor, so the situation is unreal.

As though I were...
As though you were...
As though we were...
Etc.

If I were...
If you were...
If we were...
Etc.

The sentence is fine, but that’s not the subjunctive. The subjunctive would use were, not was.

Either one is correct; were is much more formal.

Some people would go so far as to say that only were is correct, but good English speakers and writers use was in this sort of context quite naturally:

Dumbledore did not speak for a moment; he looked as though he was trying to make up his mind about something. At last he said, “I am sure. I trust Severus Snape completely.” —Rowling, J. K., Harry Potter and the Half-Blood Prince, chapter 25

As others have said, your example shows the indicative mood, not the subjunctive mood. The authors of ‘The Cambridge Grammar of the English Language’ do not even call it subjunctive, preferring the term irrealis were. They explain, persuasively enough:

Traditional grammar calls our irrealis a ‘past subjunctive’, contrasting with ‘present subjunctive’ be. But there are no grounds for analysing this were as a past tense counterpart of the be that we find in constructions like It’s vital that he be kind to her. We don’t use ‘subjunctive’ as a term for an inflectional category, but for a syntactic construction employing the plain form of the verb.

Maybe I'm wrong, but I believed the correct form was to use "were" in situations that are hypothetical, not actual.

"I wish I were a princess."

"I was tired last night, so I didn't go out."