I'm 78 and still contributing to my 401K. I'm under the impression I don't have to start withdrawing until I've quit contributing. Is this true?

Here are the rules from the IRS website.

401(k) Resource Guide - Plan Sponsors - General Distribution Rules

In general it's true that you are not required to make withdrawals until you either retire or reach age 70-1/2, whichever comes later, so in your case since you have not retired you are not required by the IRS to take distributions. However, plan rules vary from company to company, and some actually do require distributions at age 70-1/2 even if you are not retired, so check with the plan administrator to see what the rules for your plan say.