In short, practice bilingualism, not simultaneous communication.
Simultaneous communication or its term variants (SimCom and sign supported speech) is one of some "communication methods", in which one speaks both spoken language and manually coded English at the same time.
It's very commonly used among hearing signers with other hearing listeners in front of Deaf people.
Speaking two languages in two modes simultaneously may sound perfectly feasible to naive hearing people. But, it's not feasible.
Because, these two language ASL and English are entirely different with their own grammatical rules and structure as well as their own structure of the smallest units of language.
ASL has its own grammatical rule and structure that one cannot speak two languages at the same time, regardless of separate modalities. It's an analogue to speaking English and typing Arabic at the same time. Result? Broken Arabic with English-like grammar.
Not only it corrupts grammar in an oppressed language, it also loses intonation, contextual meanings (words in different contextual sentences), prosody, and all other linguistic features.
Simultaneous communication is not only unacceptable, but it's also a symbol of disrespect and history-long oppression -- audism and linguicism. Using sim-com exhibits an user's ignorance and audist behavior. It may offend Deaf people (except for a few culturally acceptable situations).
When speaking both languages at the same time, one strongly tends to, if not always, speak full English and broken ASL. What this tells is that ASL tends to be devalued and that English tends to be chosen or a priority over ASL. It explicitly exhibits an audist attitude.
Speak in one language and then another. That way one articulates full sentences in a language at a time.
Resources for more information.