Jason:
To be clear, you have until October 14 to take action to keep your case going (assuming the mailing date of April 14 is correct).
You have the following options at this time (note that ALL but #5 require a petition and fee for a three month extension of time):
1. File an RCE with an appropriate "submission" under 37 C.F.R. 1.114 (e.g., new arguments, an amendment, an information disclosure statement, etc.) Cost for the RCE is about the same as filing a new application, minus fees related to the size of the application or number of claims (assuming you don't add more claims than you previously paid for in an amendment)
2. File a notice of appeal. Costs a few hundred bucks, and will buy you 2 months (extendable up to five additional months) to decide how you want to proceed. While I can't speak to your case specifically, I usually only recommend filing a notice of appeal when: 1) the client needs extra time to perform some action (e.g., comparative testing to overcome an obviousness rejection); or 2) the client intends to actually appeal the examiner's decision to a higher power within the USPTO. If the client merely wants to file an amendment or new arguments, an RCE is usually the best route unless extra time is needed to develop the required 1.114 submission.
3. File a response after final rejection, with or without a claim amendment. I would not take this course of action if I were in your shoes, as it will not toll the response deadline and it is extremely unlikely that the examiner will have time to act on your case before the response deadline expires. FWIW, I rarely file responses after final anymore unless I can make an "apples vs. oranges" argument, or if I am presenting an amendment that will place a case in immediate condition for allowance. Otherwise, my experience has been that an examiner will respond to an response after final with an advisory action (nearly 100% of the time). And if that occurs, it will be necessary to take one of the other options mentioned in this list to properly response to the OA. Note that in you case, an advisory action would probably be fatal to the application, as it would issue after the response period expired.
4. File one or more continuing applications claiming priority to the application at issue (costs the same as filing a new application (claims and size fees included); benefits include providing substantial time to determine how to proceed with a case, opportunity to "shift" the claims to another invention, etc.
5. Abandon the application (not a desirable option in most cases).
Good luck.
Klav