Can I apply for limited company now, while fully time employed, and not take any business until I get a contract?
Some employment contracts may include non-compete clauses or similar which expressly forbid you engaging in other employment or becoming self-employed while simultaneously working for your current employer.
You may want to check this out before making any moves to register as a limited company. You may forfeit long-term benefits (such as a pension) you have built up at your present employer if they catch wind of a conflict of interest.
As noted in an earlier answer, the setup process for a limited company is extremely simple in the UK, so there is no reason you need to take these steps in advance of leaving your current employment.
During my resignation period scout for contracts...
Should I wait weeks before actually deciding to search for contracts?
Depending on the type of IT work you intend to be contracting for, you may find yourself shut out from major work if you are not VAT registered. It is a requirement to register for VAT when you breach certain earnings limits (see HMRC's website) but you can voluntarily register with HMRC before these limits if you wish.
Being VAT registered increases your bookkeeping and oversight requirements, which makes you appear more attractive to larger enterprises / corporations than a non-VAT registered firm. It also suggests some degree of stability and a plan to stick around for the long haul.
This might be a catch-22 situation - if you want to get noticed and land the sizable contracts, you will almost certainly require a VAT registration regardless of your overall yearly earnings. It would be advisable to engage the services of a professional advisor before becoming VAT registered, but this and the subsequent professional advice you may require for putting in VAT claims may not be a fee you wish to pay upfront if you are only attracting a small volume of work.