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First off, language questions are asked in the Learning Japanese subforum in the sticky made for these kinds of questions. You've been here a while, you know this.

Secondly, to say that you don't have free time the proper phrasing would be aku ga nai/arimasen(to imply you don't posses it, as opposed to just saying you aren't free)

Lastly, there are two ways to form a negative te(let's use the word presented, aku, as an example). The first one is you taking the plain negative(-nai) of a word and swapping out the last i with -kute(this form used mainly to connect sentences and such) and the second is just adding de to the plain negative form(used mainly in the construct '-nai de kudasai' to request that someone doesn't do something)

Woha, she didn't mean it like that. It's just sometimes topics that were already discussed get posted a lot and from what I understand Koiyuki is one of the only active mods right now and she doesn't want to deal with a lot of repeat threads or threads that have the same topics.

That being said, I would be more than willing to answer your question. You see when you use -te form, the ending for -te form verbs is more than likely ~te iru. Or in keigo it is ~te imasu. This usually means like a ~ing verb in English. To make any ~te form verb in Japanese negative you need to make the ending (being imasu or iru) negative, resulting in imasen (or inai in informal).

Let's see how this works out:
Postive: Akimasu (aku) to be free; open
Te-form: Aiteimasu (Aiteiru)
Negative -te form: Aiteimasen (Aiteinai)
Negative past -te form Aiteimasendeshita (Aiteinakatta)

Side note: A lot of times in speech and writing (and most J-Pop/Rock song lyrics) they shorten the -te form verbs by dropping the i of iru.

Example: Aiteru (あいてる) instead of Aiteiru (あいている). It makes the verb roll off the tongue faster.