shelter question

hi, so me and my family are four are staying in the shelter and long story short my husband got a job finally but it’s a real part-time job only 20 hours a week $16 an hour. We’re barely getting by we do have food stamps, but they said as long as we have four paystub‘s, we could get the voucher and start looking for an apartment, which is great my question is should we do it or should we wait till you find something better cause he still applying for jobs to get something full-time I don’t wanna go barely making $300 a week and we can’t cover our cost like I don’t know what they pay for. I know they help rent, but I’m sure we have to pay to keep the lights on and we have to pay our phone bills and if the kids need something or just enjoy our life in general and go out I don’t think we could do that. We’re just 200. Maybe $300 a week it’s only three days he’s working so I really want advice suggestions like I’m willing to tough it out at this place if it means, we’ll be able to have a better job and live comfortably. I’m confident he’ll find a job soon and all we need is four paystub’s and then we can get the vouchers so either way it’s a quick process, but I don’t wanna rush into it and then we drown. and we’re in queens, New York just for reference. I just really wanna know if I should take the voucher out and look for an apartment or wait till I get a better job.???