I assume that this is a nonqualified annuity, not an IRA annuity or an annuity in a qualified employer plan. You can only avoid a taxable distribution from a nonqualified annuity if you move the annuity explicitly by what is called a "1035 exchange," essentially a trustee-to-trustee transfer from one annuity to another where no funds are actually being paid to you personally, or if your investment in the contract is greater than the amount distributed and the distribution would be nontaxable even if not deposited into another annuity.
There are ways to transfer a withdrawal from one annuity to another as a 1035 exchange. There are some specific rules about what can be done so you should get some tax guidance to make sure you follow them.
Here is what investopedia says:
A 1035 exchange is a provision in the Internal Revenue Service (IRS) code allowing for a tax-free transfer of an existing annuity contract, life insurance policy, long-term care product, or endowment for another one of like kind.
I am taking money from a nonqualified account and putting into a qualified account but no with an
employer. Is this taxable with a penalty?
The age restriction for contributions to a traditional IRA has been eliminated. However, you must have income from working to contribute to a qualified account (such as an IRA). Do you?
And then you must start taking Required Minimum Distributions at a certain age.
This web article should answer your questions:
https://pocketsense.com/can-annuities-changed-ira-tax-penalty-5369.html