First question - do you actually want to cycle or this something DH has imposed on you?
Assuming it's something you want...
Do you always go out with him? Sounds like he sees himself as the better cyclist who needs to coach/instruct you. That doesn't always work well between husbands and wives/partners. Could you go out by yourself for a while so you get used to your own pace rather than trying to follow the one he sets?
Agree on raising the saddle. Can you put your foot flat on the ground while you are in the saddle? You shouldn't be able to. Tiptoe only. (In fact I can't touch the ground from my saddle at all - I hop off as I stop and back on as I start. That's not ideal - a compromise of being too small for a lot of bikes - but I mention it to highlight you really don't need a flat foot on the ground)
Would you try clips for a while? Then you can't move your foot too far forward. (Clips as in the old school cage things, no special shoes required and less practice needed to get your feet in and out of).
Do you drive? Then you know how to use gears. Think of working through the gears as you start off.
Are you pedalling fast or slow? If your feet at moving at walking pace it's too slow - drop down a gear and pedal lighter but faster. That might be why you are bobbing as well - you are having to push too hard with each stroke. You want the pedalling to be light - like once you get it up to momentum you are just keeping it spinning round not pushing on the pedals. (I used to think of Respect by Aretha Franklin - RESPECT, each one a stroke!)
Obviously in reality there will be times you pedal faster or slower, but try and make you usual cadence light and fast not heavy and slow.
Also all the above assumes you are cycling on road on a road bike or a sporty hybrid. If it's MTB then I have no idea :) and if it's a sit up and beg with a basket then you and DH are trying to do fundamentally different types of cycling.
On buying a new bike, unless there's a backstory here (suspect there may be), I'm actually not a fan of buying new kit until you've reached the limitations of your current stuff - which means not just "this doesn't work for me" but understanding why it doesn't work and therefore what you need to look for in the replacement. If your DH thinks the bike is ok it's probably ok for you for now.
That said, it might be worth (after lockdown) demoing some bikes at a bike shop anyway - that will show you quickly if it's your technique or the bike that's holding you back. Good bike shops let you book a demo and ride the bike around.