The House Ways and Means Committee announced it will begin drafting a tax bill Thursday. Meanwhile, Senate Finance Committee Chairman Lloyd Bentsen (D-Tex.) said his panel would begin its tax-writing sessions next week. The action came after President Reagan decided to sign the measure revamping the Gramm-Rudman budget-balancing law. The legislation will require that the fiscal 1988 deficit be reduced by $23 billion. If the President and Congress do not agree on ways to meet that goal, the difference will come from automatic equal cuts from domestic and military programs. Bentsen and other congressional Democrats have said about half of the deficit reduction should come from new taxes. Reagan has called for cutting domestic programs instead of the military, and promised to veto any tax boost.