I've got something to say is fine.
HAVE got can be used almost anywhere that bare HAVE is used as a "lexical" verb—that is, as a main verb rather than a perfect auxiliary. It can also replace HAVE in the "periphrastic modal" HAVE to = 'must', as in "I've got to get a haircut."
The principal restriction on HAVE got is that it can be used only with HAVE in the simple present form: I/you/we/they have got, he/she/it/John has got. It can't be cast in the past-tense form, or the progressive construction, or in the infinitive or participial form.
There are marked differences in BrE and AmE use. In AmE HAVE got is largely restricted to conversational situations, probably because it was for a long time deprecated as vulgar by US English teachers. It is acceptable in all registers in BrE.
In BrE HAVE got is routinely used in questions and negations:
Have you got any money?
I haven't got a penny.
This is rare in AmE; US speakers tend to revert to the ordinary HAVE in these contexts:
Do you have any money?
I don't have a penny.
The distinction may be attributable to the AmE preference for gotten as the past participle of get, where BrE speakers prefer got: BrE treat HAVE got as a sort of idiomatized perfect. This analysis is supported by the fact that ain't got is fairly ordinary, and by the growing frequency of bare got, without the HAVE, in AmE:
You got any spare cash? We gotta get some gas.
Yeah, I got plenty, I'll buy.
Colloquial AmE, it appears, is coming to regard got as the core of the expression and HAVE as a dispensable ornament.
The descriptions of AmE use rest on my own analysis of the UCSB corpus of spoken English. In conversational contexts I found 545 instances of indicative have/has and 158 instances of DO have, against 171 instances of HAVE got, all indicative, and 90 instances of bare got. In non-conversational contexts I found 167 instance of have/has/DO have and only 4 instances of HAVE got.
I have ridden a rollercoaster and now I am dizzy.
The only things wrong with your example sentence is that roller coaster is two words, and there should be a comma after them. This is to separate the two independent clauses. Otherwise, the grammar is fine.
Contextually, most folks would connect the two clauses as cause and effect. That is not the best way to word it, though. Potentially the only time you have ever ridden a roller coaster was 15 years ago, and you found yourself sidetracked mid-sentence (by your dizziness, perhaps). To force the interpretation you want, it would be better to say something like:
I just rode the roller coaster, and now I am dizzy.
Best Answer
Yes, absolutely. Whether it's correct is a more difficult question, and of course, language "correctness" itself is a difficult concept.
As a native speaker, I wouldn't blink at that at all and might well say it. Other ways I might say it:
I've figured it out.
I've figured it out now.
I've got it.
I've got it now.
Curiously, though, I would never say "I've figured it out now" in the context you gave, but I would in a different context, for instance if someone asked me if I still needed help: "No, thanks, I've figured it out now."