See the following cartoon:
Position the text shapes and select them, Group them to prevent accidental moves
Make a copy. Goto the Pathfinder panel and select Unite (this is not a must, but can make the handling easier)
Give to the copy the wanted fill color and a thick enough stroke with the same color
Draw a shape that fills the holes and has the wanted form. Use the pen tool to make the curves you need. This example has no artistical goals, it's only for the method
Give to the filler shape the same fill color
Drag the original shapes onto the background shapes
You can edit the background shape with the direct selection tool. If you want to make some sharp corners smooth, you can convert the corner anchor point to smooth with the anchor point type conversion tool (=drag the wanted handle lengths and directions)
INSERT due the extended question:
Here's something fast with the same technical methods:
There is a with the pen drawn black shape in the middle under the words, the outline is a circle, united with a copy of the N with thick stroke the texts have black stroke + decorative lines. The curve with dots is prepared to a brush for easy repeat. The texts were originally written with a font, but converted to curves (=Outlines) and distorted totally differently than in your example.
Inserted the full receipe:
Write the text with the ordinary text tool. The name of the font is taken from questioner's previous question.
Convert the text to curves (= Type > Create Outlines), drag to bigger size, give white fill + black stroke
Draw a circle, no fill, black stroke; the size = Give fits in the middle and Up below it
Drag i lower and draw a circle over it, drag Never much smaller
With Object > Envelope distort > Make with warp distort words Never and Up. Apply distortion Upper arc to Never and Lower arc to Up. The used bending was 50%
Expand the distorted objects (not strokes and fills, only objects), Ungroup them until the characters can be selected separately. Drag Never and Up smaller and place them. You must do separately N, ever, U and p.
Copy and paste in place N. Give to it the same thick stroke a the circle has. Remove the fill. Select the circle and the copy. Apply Unite in the pathfinder panel. The rest of the objects are temporarily blanked here only to see clearly the result.
Take the Pen tool. Draw the bottom shape and fix the curvatures & anchor point placements with the direct selection tool. It's colored and on the top here only to see it. Draw with the same stroke as the characters have. Smart quides can help to make it fit.
The bottom shape has got black fill and it's arranged to the back
Draw a thin rectangle and 3 circles, no stroke, black fill. Drag it into the brushes collection and select it to be a new pattern brush
Take the Pen tool and draw some decorative lines, apply the new brush and select thin enough stroke to each. Fine-tune the curves with the direct selection tool.