No, don't sow seed as you go. It will be messy, awkward and time consuming.
Just take your time, get the whole lot turned and levelled before sowing any seed. The most important part is to level properly......rake and level, walk on your heels. Rake again. Walk again. Get that surface firm so that bumps and hollows are removed. Get a nice tilth. A week or so before sowing scatter a general granular fertiliser and rake it in.
Choose a nice dry amd not too windy day. Sow half your seed one way. Lightly rake it in....just enough to move seed but not too much soil. Then sow seed from opposite direction. Rake again...lightly.
Water in gently.
If the area is not too big a covering of fleece will help keep off birds, pests and retain some moisture.
Sowing seed next month is good. Your lawn will grow rapidly from then.m