If you use 'I'm,' it is a contraction for 'I am' which is the present tense of the verb 'To Be.' So, your sentence has to have subject/verb agreement throughout. You need to write '...I want to go...' and 'I am stuck...'
If this was something that happened in the past, you would write that you 'wanted to go,' but you 'were stuck studying.' Just a matter of good sentence construction, you do not need 'but' and 'instead' together. EX--using the present tense: 'I want to go to Joe's party, but I am stuck studying.' OR, 'I wanted to go to Joe's party; instead, I am stuck studying.'