There is no reason to remove the old mulch, you can simply put new on top of the old. You will probably find that most of the "weeds" are growing in that old mulch and you probably could remove the "weed" blocking fabric although even that is not totally necessary.
How long and material you use for mulch depends on what the muclh is and how active the Soil Food Web is. Many people with dead soils find that wood chip mulches last several years while others of us with a living soil find that even the wood chip mulches can be digested in a year.

My condolences on your loss.

You're dealing with the reason I hate landscape fabric. In due time the mulch on top becomes soil and the weeds grow anyway. People don't think long-term about it.

I'd be inclined to rip it up and get rid of it. If the mulch is deep enough and tended often enough to work as mulch its deep enough without the cloth.

If I had to put something under the mulch to smother sod I'd go with paper or cardboard.

Thanks everyone. I'll just toss some new mulch on top then. In my own home I don't have landscaping fabric and the old mulch just blends right in. At this point I'm not seeing how the landscaping fabric is saving me much. Initially it did but in the long run I'm in the same place.

It more depends on whether the gardens are in good shape as-is and if you are happy with the plantings and planning to leave everything as-is, versus are having trouble with dying shrubs, intend to plant new shrubs or annuals from time to time, in which case you would be more interested in long-term gardening issues, such as improving soil content and structure being able to move things about from time to time, and the fabric interferes with that.

If not doing anything but keeping it neat and tidy, then just more mulch is simplest.

The people at Texas A & M have found that the landscape fabrics can slow the movement of nutrients into the soil. Plots with landscape fabric had lower levels of nutrients in the soil then plots with no landscape fabric even when the same amount of "fertilizer" was added.
Mulches are put down to 1) aid in control of unwanted, "weed", plant growth, 2) aid in soil moisture retention, 3) aid in soil temperature control, and 4) add organic matter to soil. So mulches are meant to be digested and added to soil, over time, just as Ma Nature does in the forests of the world.