sectioning format

I would like to achieve the following look:

        01. Section 1 Name
        1. Text in a numbered paragraph …
           Text in a new, unnumbered paragraph …

        2. Text in a numbered paragraph …
           Text in a new, unnumbered paragraph …

I get it when I enter the paragraph numbers by hand in:

        \begin{document}
                \section{Section 1 name}
                        \paragraph{1.}
                                Text in this numbered paragraph …

                                Text in a new, unnumbered paragraph …

                        \paragraph{2.}
                                Text in this numbered paragraph …

                                Text in a new, unnumbered paragraph …

        \end document

The following does not number the required paragraphs at all.

        \begin{document}
                \section{Section 1 name}
                        \paragraph{}
                                Text in this numbered paragraph …

                                Text in a new, unnumbered paragraph …
                        \paragraph{}
                                Text in this numbered paragraph …

                                Text in a new, unnumbered paragraph …
        \end document

When I add in the preamble

                \setcounter{secnumdepth}{4}

the paragraphs do get numbered but as follows:

        01. Section 1 Name
        0.1.0.0.1  Text in a numbered paragraph …
           Text in a new, unnumbered paragraph …

        0.1.0.0.2  Text in a numbered paragraph …
           Text in a new, unnumbered paragraph …

Is there a way to get rid of the leading 0.1.0.0.?
Companion2ed, bottom page 25 says "Thus, by changing the counter 
representation commands, it is possible to change the number displayed 
by a sectioning command." I tinkered a bit from the examples, including 
with \@parcntformat, but I think that it is still beyond what I can do 
at this point.

Also, I wouldn't mind getting rid of the leading 0 in

        01. Section 1 Name

Regards
--schremmer

P. S. I had used asparaenum to obtain that effect, which was clunky but 
seemed to give me what I wanted, but I just realized that
        Text in a new, unnumbered paragraph …
is not treated the same way after
        \begin{asparaenum}
        \item
as after \section. Hence my attempt to use \paragraph instead.
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