You misread the page about sniffing packets; it is already paranoid and alarmist, but not to that point.
When producing a DSA signature, the signature engine works in a group which is embodied by a modulus p (a big prime number) and a group order q (a smaller prime, divisor of p-1). These values are part of the public key. For each signature, the signature engine must use a random number generator to produce a random value k which MUST be chosen uniformly in the 1..q-1 range. The "uniformly" term is very important. If the choice of k is biased, then it allows for key recovery after observing a number of signatures. Whether this is critical or not depends on the bias. A relatively small bias (e.g. k is generated as a sequence of bits of the size of q, then reduced modulo q) will incur an "academic" break with key recovery after about 263 signatures: an attack, yes, but not practical. On the other hand, if the random generator is so botched that it always outputs the same value for k, then two signatures are sufficient to recompute the private key. Sony deployed such an utterly failed RNG/signing system.
What this amounts to is that if the implementation of a signature algorithm is bogus, well, it may leak important data, including the private key. This is true of any cryptographic algorithm, and RSA is not exempt. Singling out DSA is unwarranted.
(What can be said in all neutrality is that DSA and ECDSA require a source of randomness for each signature, which is a relatively heavy requirement for embedded systems such as smartcards. Solutions exist; when I have time, I work on the issue. Edit: this is now a published RFC.)
As for key size, the current standard specifies sizes of 1024, 2048 and 3072 bits (that's the size of p). An older version of the standard allowed only 1024 bits, while an even older allowed all sizes multiple of 64 from 512 to 1024 bits. A number of deployed implementations of DSA are a bit lagging behind the 2009 standard, and thus allow only 1024-bit keys.
A 1024-bit DSA key is not a critical risk. As far as we know, DSA appears to be at least as strong as RSA with the same key size, and in practice a bit stronger. DSA relies on discrete logarithm and the current DL-break record is for 530 bits, while the current record for factorization (RSA) is 768 bits. Best known algorithms for factorization and DL have a lot of similarities, but the final step (where a lot of fast RAM is needed) appears to be more expensive with DL than with factorization. Thus, right now, a 1024-bit DSA key is not a true security risk. Especially since, for signatures, we have less to worry about future break than for encryption; and DSA is signature-only.
The main issue with DSA is that it is not as widely supported as RSA; when it is supported, it may have limitations (e.g. limited to 1024-bit keys, or usable in conjunction of only a limited set of other cryptographic algorithms). These limitations are not intrinsic to the algorithm, but reflect the state of the market, which is dominated by RSA. However, it is currently quite fashionable to switch to elliptic curve variants of DSA and, correspondingly, the market is slowly (very slowly) shifting towards more support for (EC)DSA.