Ok, They replied, but they seem to think I am a Philippines citizen. I am also trying to get my notarized stuff from Seattle right now, so I will ask them when they call me about it. Here's what the Philippines office said (my comments in brackets):
1. First you must leave Taiwan before your current ARC expired.
2. The validity of Student Resident Visa is 3 months, but you should apply a new ARC within 15 days upon your arrival in Taiwan. The Immigration Office may not agree to grant you student ARC before the start day of school.
3. If you intend to apply a visit visa to Taiwan between July 13th and September, you should leave Taiwan when the said visit visa due, then apply the student visa to re-enter Taiwan.
[So a visitor visa won't work, it will just buy time until the next visa run, but I am pretty confident they will give me the student visa]
RESIDENT VISA REQUIREMENTS
FOR STUDYING IN COLLEGE OR UNIVERSITY IN ROC
1. An online Visa Application Form (please access https://visawebapp.boca.gov.tw
to fill up the Form and print them out)
2. Two (2) 2 X 1.5 head & shoulder photos against white background (taken w/in (3) months)
3. Passport valid for at least six months and old passport showing previous travels
4. Birth Certificate issued by NSO [I am not a Philippines citizen so I replied to ask if this is necessary]
5. Marriage Contract (if applicable) issued by NSO
6. NBI Clearance [Equivalent of an FBI background check. Again, I hope this isn't necessary for U.S. citizens]
7. Diploma and Transcript of Records ( if issued by Philippines school should be authenticated by Commission on Higher Education (CHED), Department of Foreign Affairs (DFA) and TECO)
8. Admittance letter from the accredited schools
9. Medical Examination Clearance (valid only for three months from date of issue) from any of the following accredited Hospitals and Clinics : [A bunch of places in the Philippines]
10. Financial supporting documents of the applicant such as:
（2）Bank Book or Bank Statements
（3）Financial Support Letter from parents or relatives (Affidavit should be notarized) [To cover my bases, I'm bringing all three]
11. The Student ID card. [Don't know if this will be possible for me]
13. All documents submitted to this office should be Original and 1 photocopy
NOTE: THIS OFFICE RESERVES THE RIGHT TO WITHHOLD THE REASON FOR THE DISAPPROVAL OF ANY VISA APPLICATION AND THE FEE IS NON-REFUNDABLE