As Brian pointed out, there's no need to include the 的 part the way you structured your sentence. However in order to sound a little more natural, you need to say something like this:
我太高兴昨天去旅游的时候吃了晕车药/晕船药,因为我晕车/晕船。
In this case the 是。。。的 phrase is not very suitable, as Beth said, use 了 instead.
Notice we don't have this Motion sickness phrase in Chinese, you need to be more specific of what you are traveling with.
Also avoid using the phrase 是吃了药的 instead you should use 吃药了吗/吃过药了。If you say somebody 是吃了药的,it sounds like he/she is mentally challenged and is on something.
@Beth, the 是。。。的 phrase does not highlight the circumstances of past events, you can use it to describe current circumstances too. 这些人是有钱的,那些人是没钱的。这些人是来工作的,那些人是来玩的。
Finally, just before I shut my month, avoid using dependent clause in Chinese as you'd do in English. If you absolutely have to use one, reverse the order of them in 99% of the cases will make your sentence more natural.
I ate the cake that my wife got me yesterday.
In Chinese we don't say things in this order normally, instead we say:
我把昨天我老婆给我买的蛋糕吃了。one sentence and that's it.
If you say: 我把蛋糕吃了,就是昨天我老婆给我买的那个,then it gives the impression that you realized right after your finished the first part (我把蛋糕吃了) of the sentence that you were not specific enough and therefore need to add the second part (就是昨天我老婆给我买的那个) after it in order to avoid people asking you "what cake?"
Sorry about this long novel, hope it helps, good luck with your Chinese.