If you wish to marry in the Church, an annulment is required. If you even think it is a possibility, it's better to go through the process sooner rather than later: if you have to go through the full process it can take a fair amount of time, which will seem like an eternity if you are holding off on a relationship waiting for it.
Yes Jerry, if you are in a relationship and are deciding for a church wedding, that will speed up the process.
The pressing circumstances may get the case moved to the top of the stack and started sooner; however, the investigations, waiting for responses, interviews, etc. take time -- potentially a lot of time, depending on the circumstances. Showing up at the last minute and crying isn't going to change any of that.