I know exactly what you are dealing with... the IC is the famous TX-2B / RX-2B or TX-2C / RX-2C chip set... - google 4 or 5 channel remote control IC 27MHz (images and then sites) where you find plenty of information... including some details of the drivers for the motor and solenoids, for example, this with variations.
As I see from your pic, there is a plenty of transistors/fets on the board that drive the things... you can reverse engineer, but you also can go just by yourself. What I suggest is to get a quad half-bridge chip, where with one you drive the motor, and with the other you drive the solenoids in series, for example something like SN754410 Quadruple Half-H Driver running on 5V, but able to run on 3.7V: Allegor A3988 quad half bridge drivers.
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I know exactly what you are dealing with... the IC is the famous TX-2B / RX-2B or TX-2C / RX-2C chip set... - google
4 or 5 channel remote control IC 27MHz
(images and then sites) where you find plenty of information... including some details of the drivers for the motor and solenoids, for example, this with variations.As I see from your pic, there is a plenty of transistors/fets on the board that drive the things... you can reverse engineer, but you also can go just by yourself. What I suggest is to get a quad half-bridge chip, where with one you drive the motor, and with the other you drive the solenoids in series, for example something like SN754410 Quadruple Half-H Driver running on 5V, but able to run on 3.7V: Allegor A3988 quad half bridge drivers.
Since you do not have a lot of space, you may salvage the drivers from your current board - if they are still any good. Otherwise, check out some examples for H-Bridges with bipolars or examples for H-Bridges with FETs. You may find either on your current board...