It is no way safer using a full node as wallet than using a lightweight wallet like electrum. You never need to trust 'an external server' when broadcasting the TX.
You can always push your transaction yourself. But you have to relay it to other nodes. How you will accomplish this.. is up to you.
Has nothing to do with running a full node for security reasons.
SPV wallets does have issues regarding the privacy and security. In the privacy aspect, servers are able to see all the addresses associated with an IP address or a wallet. Regarding the security, SPV assumes that the longest chain difficulty-wise is valid. Since it doesn't validate the rules, SPV clients are at risk if there is a fork due to blocks not following the rules that majority of the network is following. This has happened before and the repercussions can be serious.You can always push your transaction yourself. But you have to relay it to other nodes. How you will accomplish this.. is up to you.
Has nothing to do with running a full node for security reasons.
Electrum is better than most SPV wallets in this respect, though. Versions 2.9 and later have a fork detection mechanism if there is a fork dispute among servers. Users are then prompted to choose which fork to interact with.
You also don't need to randomly connect to servers with Electrum, so you won't necessarily follow the most-work chain if it's invalid. You can connect to a Core node server, removing this concern. For example, here is a list of Electrum servers. You can choose one that is running the Satoshi client.
There are definitely privacy weaknesses. But there are good reasons why Electrum is listed on bitcoin.org -- because it's secure, open-sourced, and miles ahead of all other SPV implementations (usability, etc).
