Hi harmony1
Yes it is correct, we always advise claimants to apply for UC just after their ESA has gone into their bank a/c, you have to get the timing right.
UC is paid monthly, from the information you have supplied I would say your assessment period is 27th June to 26th July, you then get paid 7 days later 2nd August.
What many claimants do is to ask for an advanced payment, to tie them over. The advanced payment will be recovered from subsequent UC payments, usually within a maximum of 24 months.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems