Yodrak

Pammy,
If you qualify for a travel document, 'I want to' is a sufficient reason for one to be granted. 'I might want to' is even good enough for advance parole for a K1 adjustment applicant.
If you're going to be gone for a long period of time be sure that you take steps to preserve the USA as your primary place of residence that you do not intend to abandon. Time alone - 1 year - will result in a presumption of abandonment for an LPR who has not obtained a re-entry permit.
Yodrak

All,
I-134 is not I-864, and a non-immigrant is not an immigrant. There is no requirement that the petitioner for a K visa must be the sponsor for a K-visa applicant to meet the public charge provisions of the law. An applicant who has sufficient financial resources of their own does not need a sponsor. However, most applicants do need a sponsor and the most reasonable person to be the sponsor, when one is needed in the case of a K visa, is the petitioner.
Whoever the sponsor will be, they must provide financial documentation to show that they can support the visa applicant while the visa applicant is in the USA. This need not necessarily include an I-134, which is optional at the discretion of the consular officer processing the case, although an I-134 is almost always asked for.
Some K-visa applicants in the UK have found that they did not sponsorship information from their petitioner when they offered someone else as the sponsor. But the anecdotal evidence is very sparse. It would be prudent, when someone other than the petitioner is put forth as the sponsor, to have the petitioner's financial information on hand in the event that it is asked for.
Yodrak

gracy_byron,
I-797 (and I-797C) is a paper with a fancy background on which the USCIS prints the Notice of Action. NOAs come in many flavors, the most loved being Receipt Notice (aka NOA1) and Approval Notice (aka NOA2).
Yodrak

All,
I-134 is not I-864, and a non-immigrant is not an immigrant. There is no requirement that the petitioner for a K visa must be the sponsor for a K-visa applicant to meet the public charge provisions of the law. An applicant who has sufficient financial resources of their own does not need a sponsor. However, most applicants do need a sponsor and the most reasonable person to be the sponsor, when one is needed in the case of a K visa, is the petitioner.
Whoever the sponsor will be, they must provide financial documentation to show that they can support the visa applicant while the visa applicant is in the USA. This need not necessarily include an I-134, which is optional at the discretion of the consular officer processing the case, although an I-134 is almost always asked for.
Some K-visa applicants in the UK have found that they did not sponsorship information from their petitioner when they offered someone else as the sponsor. But the anecdotal evidence is very sparse. It would be prudent, when someone other than the petitioner is put forth as the sponsor, to have the petitioner's financial information on hand in the event that it is asked for.
Yodrak

Kinsey,
The consular officer has your original approved petition. They don't need you to give them a copy of it.
Your fiance should know what information you provided in the petition in case he is asked a similar question during the interview. But that is the kind of basic information that he should know about you anyway, he shouldn't have to learn it from seeing a copy of the petition.
Yodrak

It dawns on me that with the new fee structure a K3 adjustment applicant has a tougher decision to make about when to apply for employment authorization.
Presently there is a potential benefit to a K3 for applying for EA prior to applying for adjustment - the EA will last until the authorized stay in the USA ends, 2 years from entry. Twice the time for the same amount of money and no hassle of renewing should the adjustment processing take more than a year.
Under the new fee structure a K3 who wants the longer period of EA will have to pay a significant cost penalty for it - a tough call in districts where AOS process may take about a year, and worth avoiding the hassle of renewal or not in districts where AOS is sure to take longer than a year?
Yodrak