It could be one sentence, probably, but that’d be very convoluted. You’d probably end up with something like 病気なのですごく悲しくて次の月曜日に空港に行くつもりだったけど行かない (and I’m pretty sure that implies some sequentiality and/or relationship between different clauses that doesn’t actually exist). Much better to split it up into at least two sentences.
A few points to note:
私は isn’t wrong, but it’s also not necessary to specify it. For lack of a different topic/implied subject, the reader will infer this, so it can be omitted.
You have the order of から wrong - the cause comes before it, the result comes after. すごく悲しいから病気だ means you’re sick because you’re very sad, you want to say 病気だからすごく悲しい.
Next monday would be 次の月曜日. 来 isn’t a universal prefix for “next …”, and in this case it becomes two units: 来月 and 曜日. You’re essentially saying you’re going to the airport on a weekday next month. I went with 空港に行く instead.
Also, 空港で出た means you left the airport, not that you went to the airport. It’s also past tense which doesn’t make sense with the specification of next monday. This particular past tense in English isn’t really a past tense in the normal sense, it’s more of a way to express that the plan was to do something (but presumably that plan may now have changed). For that reason I went with つもりだった, making it so it’s about your plan to go to the airport which is now in the past, not the actual going to the airport.
Similarly, 今 is a specification of time for the action you describe, whereas the “now” in the English version is to indicate the contrast between what the plan used to be and what it is now. I’d opt to keep this simple and say 行かない and use けど or something like it at the end of the preceding sentence to indicate the contrast. Also like above, I opted for 行く instead of 出る, for the same reasons.
So I’d say a somewhat natural way of saying this is. 病気だからそごく悲しい。次の月曜日に空港に行くつもりだったけど、行かない。
If, however, the reason you’re sad is not so much because you’re sick but because you can’t go to the airport, you can combine those into essentially “I’m very sad because I planned to go to the airport but I can’t due to being sick” - which could be something like 次の月曜日に空港に行くつもりだったけど病気で行かなくてすごく悲しい。