What sort of "full disability" are you referring to? The answers to your questions will change, depending on whether you're talking about Social Security Disability or some form of Long Term Disability insurance.
If you're talking about Social Security only, then yes, you can file while out on FMLA. You also don't need to wait until you settle your accident case. Any settlement you receive on that will not affect your Social Security Disability Insurance benefits.
You must produce medical evidence that you will be disabled for a period of twelve months to get on Social Security Disability or SSI. You should go ahead and file if you will be out of work for a year or longer, as it can take a while to be approved on a new injury, especially if you are turned down at first. If you have an attorney for your accident claim, you should consult with him or her on this, but if not you might want to get a referral from NOSSCR's website for your area. I handle Social Security Disability and SSI claims and am in Dothan and Enterprise.
If you are talking about Social Security Disability then you can file anytime you are out of work. However, your settlement COULD affect your benefit depending upon which social security disability program you ultimately get approved. If you qualify for SSDI, the benefit is based upon your previous work and a settlement would not affect your benefits. However, if you don't have enough work credits you may qualify for SSI which is based upon your financial situation. Therefore, if you get SSI and then get a lump sum from a settlement, you will no longer qualify for SSI. You need to talk to a Social Security Disability attorney and let them review your case and advise you.