Could I be using proof by contradiction too much?

One general reason to avoid proof by contradiction is the following. When you prove something by contradiction, all you learn is that the statement you wanted to prove is true. When you prove something directly, you learn every intermediate implication you had to prove along the way.

More explicitly, if you want to prove that $p \Rightarrow q$ by contradiction, you assume $p$ and $\neg q$ and derive a contradiction. None of the intermediate implications along the way can be reused because your premises were contradictory.

If you want to prove that $p \Rightarrow q$ directly, say by proving that $p \Rightarrow p_1$ and $p_1 \Rightarrow p_2$ and so on until $p_n \Rightarrow q$, then you've also proven that $p_i \Rightarrow p_{i+1}$ for all of the relevant $i$. Many of these statements might be more useful than the original statement you were trying to prove.

Another general reason to avoid a proof by contradiction is that it is often not explicit. For example, if you want to prove that something exists by contradiction, you can show that the assumption that it doesn't exist leads to a contradiction. But this doesn't necessarily give you a method for constructing the actual thing, which you might learn more from trying to do.

A third reason is that frequently, or so it seems to me, a proof by contradiction is really a proof by contrapositive, where you assume $\neg q$ and derive $\neg p$. This feels like a proof by contradiction except that you never make use of the hypothesis $p$ except at the very end, and pretending that these are proofs by contradiction will make you blind to the fact that any intermediate implications you prove in a proof by contrapositive are still valid.

In my opinion the proof by contradiction is a bad habit, when there is a direct proof. I always have the feeling that proofs by contradiction aren't so elegant, as it is necessary to read them several times, to see how someone got the idea that the contradiction will work.

A good proof does not only prove something but gives a way. In a direct proof you always know where you are, at a proof by contradiction there is at the end the contradiction which is more or less easy to see.

Let me quickly summarize what you can find in some of the other answers, then I will throw my two cents on top of that:

• Proof is a proof, as long as it is sound. Maybe something cannot be proved by contradiction but otherwise it is just as good as direct proof.
• Somehow many feel that Reducio Ad Absurdum is less elegant, or less intuitive.

Now my two cents. Please take a look at the following Wikipedia articles:

• Intuitionistic Logic
• Law of Excluded Middle
• Double Negation Elimination
• Mathematical Constructivism

As you can see there is a branch of mathematical logic, called Intuitionistic Logic, which does not accept the Law of Excluded Middle as an axiom, and because of this the Double Negation Elimination can not be used there. These assumptions lead to a branch of mathematics which – they say – more down-to-earth and more intuitive (see mathematical constructivism, constructive set theory, etc). Take for example the below quotation from the Intuitionistic Logic page:

Constructive logic is practically useful because its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that if you have a constructive proof that an object exists, you can turn that constructive proof into an algorithm for generating an example of it.

— EDIT: removed misleading paragraph about Cantor's diagonal argument, based on comments —

So to summarize, by going with an existential proof you satisfy even the intuitionistic requirements, therefore it is a reasonable argument to prefer it over a proof by contradiction, which relies on further assumptions (i.e. Law of Excluded Middle).