Structuring sentences without using the word "but"

"But" expresses ideas in opposition, not why they are in opposition, and just substituting though, although, and similar words won't change that. Your sentence could be recast as

Before resorting to chicken, I want to see how far I can get with alligator

(I might end up cooking the chicken) or

Though there was chicken in the fridge, I had to try the 'gator.

(Both meats are available, and I'm ignoring the more common one.)

I think the reason for the use of "but", or "though" etc., is that the sample phrases you gave all start with a statement and are then qualified or evaluated, as if you are defending or excusing or justifying a decision. You may try phrases where you don't justify the action you are taking.

I haven't tried it yet, but I think it should work.

compared to

This should work. I'll try it soon.

or

It's probably deeply rooted in my style of thought, but I'm starting to get bored of seeing myself use this structure so often.

compared to

I'm starting to get bored of seeing myself using 'but' all the time. I wonder why I use it so much. It must be deeply rooted in my style of thought.

So, when you come across a "but" in your writing, try to put the "but" phrase at the beginning of the sentence or make it a sentence in its own right. Then analyse what you had before the "but" and see how that relates to the statement and how it can be connected.

I want to see how far I can get with alligator. If I'm not happy with that, I can still use chicken.

No need to justify why you want to start with alligator. Be bold. You don't have to justify your decisions.

You could try this.

"I could easily resort to chicken, however, I want to see how far I can get with alligator

or even

"Easily, I could resort to chicken, although I want to see how far I can get with alligator.

This answer still seems weak to me, I feel like I'm missing something.