New-style ESA is not means-tested (in other words, your earnings or savings are not counted) so you should apply for that now, you can get it backdated for up to 3 months so you should ask for it to start from the day after your last day at work. For example, if your last day was 7th Feb you should ask for ESA from 8th Feb, even if you haven't had your final pay yet.
For the first 13 weeks (off the top of my head) it's the "assessment period" which means you will get the basic rate of ESA, £73.10/week. After that if you're put in the support group the amount will increase.
How much pay are you expecting on 29th March? If it's only a small amount you might still be entitled to some UC, even if you've applied before that.
Another date to consider is when is your next tax credit payment due? As tax credits are paid every 4 weeks it can be helpful to apply for UC just after you've received tax credits.
Whenever you apply, you should be able to request an advance payment, regardless of any savings that you have.
In general I would advise someone who is eligible for UC to apply ASAP (as you can't backdate UC so while you have not yet claimed you're missing out) but in your case it's a bit more complicated with the final wage payment coming in. I would want more info to do a calculation and confirm exactly when to apply (sorry I should have already said I'm a benefits adviser!)
Either way you should apply for new style ESA straight away and that's paid every 2 weeks so that should tide you over while you're waiting to claim UC and/or waiting for the first payment.
If you want more advice and reassurance it might be a good idea to visit citizens advice and take payslips etc so they can do calculations for you.