Yes, such a wallet exists, though it is still in beta: AtomicDEX (desktop version only) supports Namecoin (NMC).
You can even use the built-in “atomic swap” feature to trade your NMC, for example, for Komodo (KMD), which you can hold in the wallet and collect rewards or sell on many popular exchanges like Binance (affiliate link). Make sure you read the instructions first (found under Support). The atomic swap protocol requires that the maker and taker stay online and monitor the blockchains involved.
Update February 28, 2021:Dogecoin Core 1.14.3 has been released and significantly improves the synchronization speed. The method described in the original post below is no longer be necessary.
This article assumes that you have backed up your wallet and updated to the most recent version of Dogecoin Core (currently 1.14.2).
Once synchronized, consider forwarding port 22556 to help the network. This should give you over a hundred connections instead of the usual 8:
This will, of course, create significant network activity as can be seen in Dogecoin Core under Help > Debug window > Network traffic. This is my traffic from just two days:
Edit 2: I do not recommend using the full node wallet (version 1.14.2) if you’re just getting started with Dogecoin. Consider using a simple mobile wallet like Trust Wallet.